ChapterP Prerequisites Fundamental Concepts of Algebra
What can algebra possibly have to tell me about
the skyrocketing cost of a college education?
student-loan debt?
my workouts?
the effects of alcohol?
the meaning of the national debt that is more than $25 trillion?
time dilation on a futuristic high-speed journey to a nearby star?
ethnic diversity in the United States?
the widening imbalance between numbers of women and men on college campuses?
This chapter reviews fundamental concepts of algebra that are prerequisites for the study of college algebra. Throughout the chapter, you will see how the special language of algebra describes your world.
How would your lifestyle change if a gallon of gas cost $9.15? Or if the price of a staple such as milk was $15? That’s how much those products would cost if their prices had increased at the same rate college tuition has increased since 1980. (Source: Center for College Affordability and Productivity) In this section, you will learn how the special language of algebra describes your world, including the skyrocketing cost of a college education.
Algebraic Expressions
Algebra uses letters, such as x and y, to represent numbers. If a letter is used to represent various numbers, it is called a variable. For example, imagine that you are basking in the sun on the beach. We can let x represent the number of minutes that you can stay in the sun without burning with no sunscreen. With a number 6 sunscreen, exposure time without burning is six times as long, or 6 times x. This can be written but it is usually expressed as 6x. Placing a number and a letter next to one another indicates multiplication.
Notice that 6x combines the number 6 and the variable x using the operation of multiplication. A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression. Here are some examples of algebraic expressions:
Many algebraic expressions involve exponents. For example, the algebraic expression
approximates the average cost of tuition and fees at public U.S. colleges for the school year ending x years after 2000. The expression means and is read “x to the second power” or “x squared.” The exponent, 2, indicates that the base, x, appears as a factor two times. The negative sign in front of indicates that is multiplied by .
Exponential Notation
If n is a counting number (1, 2, 3, and so on),
is read “the nth power of b” or “b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. The expression is called an exponential expression. Furthermore,
For example,
Objective 1: Evaluate algebraic expressions
Evaluating Algebraic Expressions
Objective 1Evaluate algebraic expressions.
Watch Video
Evaluating an algebraic expression means to find the value of the expression for a given value of the variable.
Many algebraic expressions involve more than one operation. Evaluating an algebraic expression without a calculator involves carefully applying the following order of operations agreement:
The Order of Operations Agreement
Perform operations within the innermost parentheses and work outward. If the algebraic expression involves a fraction, treat the numerator and the denominator as if they were each enclosed in parentheses.
Evaluate all exponential expressions.
Perform multiplications and divisions as they occur, working from left to right.
Perform additions and subtractions as they occur, working from left to right.
An equation is formed when an equal sign is placed between two algebraic expressions. One aim of algebra is to provide a compact, symbolic description of the world. These descriptions involve the use of formulas. A formula is an equation that uses variables to express a relationship between two or more quantities.
Here are two examples of formulas related to heart rate and exercise.
The process of finding formulas to describe real-world phenomena is called mathematical modeling. Such formulas, together with the meaning assigned to the variables, are called mathematical models. We often say that these formulas model, or describe, the relationships among the variables.
Example 2 Modeling the Cost of Attending a Public College
The bar graph in Figure P.1 shows the average cost of tuition and fees for public four-year colleges, adjusted for inflation. The formula
models the average cost of tuition and fees, T, for public U.S. colleges for the school year ending x years after 2000.
Use the formula to find the average cost of tuition and fees at public U.S. colleges for the school year ending in 2020.
By how much does the formula underestimate or overestimate the actual cost shown in Figure P.1?
Because 2020 is 20 years after 2000, we substitute 20 for x in the given formula. Then we use the order of operations to find T, the average cost of tuition and fees for the school year ending in 2020.
Use the formula described in Example 2, to find the average cost of tuition and fees at public U.S. colleges for the school year ending in 2016.
By how much does the formula underestimate or overestimate the actual cost shown in Figure P.1?
Sometimes a mathematical model gives an estimate that is not a good approximation or is extended to include values of the variable that do not make sense. In these cases, we say that model breakdown has occurred. For example, it is not likely that the formula in Example 2 would give a good estimate of tuition and fees in 2050 because it is too far in the future. Thus, model breakdown would occur.
Objective 2: Use mathematical models
Formulas and Mathematical Models
Objective 2Use mathematical models.
Watch Video
An equation is formed when an equal sign is placed between two algebraic expressions. One aim of algebra is to provide a compact, symbolic description of the world. These descriptions involve the use of formulas. A formula is an equation that uses variables to express a relationship between two or more quantities.
Here are two examples of formulas related to heart rate and exercise.
The process of finding formulas to describe real-world phenomena is called mathematical modeling. Such formulas, together with the meaning assigned to the variables, are called mathematical models. We often say that these formulas model, or describe, the relationships among the variables.
Example 2 Modeling the Cost of Attending a Public College
The bar graph in Figure P.1 shows the average cost of tuition and fees for public four-year colleges, adjusted for inflation. The formula
models the average cost of tuition and fees, T, for public U.S. colleges for the school year ending x years after 2000.
Use the formula to find the average cost of tuition and fees at public U.S. colleges for the school year ending in 2020.
By how much does the formula underestimate or overestimate the actual cost shown in Figure P.1?
Because 2020 is 20 years after 2000, we substitute 20 for x in the given formula. Then we use the order of operations to find T, the average cost of tuition and fees for the school year ending in 2020.
Use the formula described in Example 2, to find the average cost of tuition and fees at public U.S. colleges for the school year ending in 2016.
By how much does the formula underestimate or overestimate the actual cost shown in Figure P.1?
Sometimes a mathematical model gives an estimate that is not a good approximation or is extended to include values of the variable that do not make sense. In these cases, we say that model breakdown has occurred. For example, it is not likely that the formula in Example 2 would give a good estimate of tuition and fees in 2050 because it is too far in the future. Thus, model breakdown would occur.
Objective 3: Find the intersection of two sets
Sets
Objective 3Find the intersection of two sets.
Watch Video
Before we describe the set of real numbers, let’s be sure you are familiar with some basic ideas about sets. A set is a collection of objects whose contents can be clearly determined. The objects in a set are called the elements of the set. For example, the set of numbers used for counting can be represented by
The braces, { }, indicate that we are representing a set. This form of representation, called the roster method, uses commas to separate the elements of the set. The symbol consisting of three dots after the 5, called an ellipsis, indicates that there is no final element and that the listing goes on forever.
A set can also be written in set-builder notation. In this notation, the elements of the set are described but not listed. Here is an example:
The same set written using the roster method is
If A and B are sets, we can form a new set consisting of all elements that are in both A and B. This set is called the intersection of the two sets.
Definition of the Intersection of Sets
The intersection of sets A and B, written is the set of elements common to both set Aand set B. This definition can be expressed in set-builder notation as follows:
Figure P.2 shows a useful way of picturing the intersection of sets A and B. The figure indicates that contains those elements that belong to both A and B at the same time.
If a set has no elements, it is called the empty set, or the null set, and is represented by the symbol Ø. Here is an example that shows how the empty set can result when finding the intersection of two sets:
Objective 5: Recognize subsets of the real numbers
The Set of Real Numbers
Objective 5Recognize subsets of the real numbers.
Watch Video
The sets that make up the real numbers are summarized in Table P.1. We refer to these sets as subsets of the real numbers, meaning that all elements in each subset are also elements in the set of real numbers.
Notice the use of the symbol ≈ in the examples of irrational numbers. The symbol means “is approximately equal to.” Thus,
We can verify that this is only an approximation by multiplying 1.414214 by itself. The product is very close to, but not exactly, 2:
Not all square roots are irrational. For example, because Thus, is a natural number, a whole number, an integer, and a rational number
The set of real numbers is formed by taking the union of the sets of rational numbers and irrational numbers. Thus, every real number is either rational or irrational, as shown in Figure P.4.
Figure P.4 Every real number is either rational or irrational.
The set of real numbers is the set of numbers that are either rational or irrational:
The symbol ℝ is used to represent the set of real numbers. Thus,
Example 5 Recognizing Subsets of the Real Numbers
Consider the following set of numbers:
List the numbers in the set that are
natural numbers.
whole numbers.
integers.
rational numbers.
irrational numbers.
real numbers.
Solution
Natural numbers: The natural numbers are the numbers used for counting. The only natural number in the set is because (9 multiplied by itself, or is 81.)
Whole numbers: The whole numbers consist of the natural numbers and 0. The elements of the set that are whole numbers are 0 and
Integers: The integers consist of the natural numbers, 0, and the negatives of the natural numbers. The elements of the set that are integers are and
Rational numbers: All numbers in the set that can be expressed as the quotient of integers are rational numbers. These include and Furthermore, all numbers in the set that are terminating or repeating decimals are also rational numbers. These include and 7.3.
Irrational numbers: The irrational numbers in the set are and Both and π are only approximately equal to 2.236 and 3.14, respectively. In decimal form, and π neither terminate nor have blocks of repeating digits.
Real numbers: All the numbers in the given set are real numbers.
The real number line is a graph used to represent the set of real numbers. An arbitrary point, called the origin, is labeled 0. Select a point to the right of 0 and label it 1. The distance from 0 to 1 is called the unit distance. Numbers to the right of the origin are positive and numbers to the left of the origin are negative. The real number line is shown in Figure P.5.
Real numbers are graphed on a number line by placing a dot at the correct location for each number. The integers are easiest to locate. In Figure P.6, we’ve graphed six rational numbers and three irrational numbers on a real number line.
Every real number corresponds to a point on the number line and every point on the number line corresponds to a real number. We say that there is a one-to-one correspondence between all the real numbers and all points on a real number line.
Objective 5: Recognize subsets of the real numbers
The Set of Real Numbers
Objective 5Recognize subsets of the real numbers.
Watch Video
The sets that make up the real numbers are summarized in Table P.1. We refer to these sets as subsets of the real numbers, meaning that all elements in each subset are also elements in the set of real numbers.
Notice the use of the symbol ≈ in the examples of irrational numbers. The symbol means “is approximately equal to.” Thus,
We can verify that this is only an approximation by multiplying 1.414214 by itself. The product is very close to, but not exactly, 2:
Not all square roots are irrational. For example, because Thus, is a natural number, a whole number, an integer, and a rational number
The set of real numbers is formed by taking the union of the sets of rational numbers and irrational numbers. Thus, every real number is either rational or irrational, as shown in Figure P.4.
Figure P.4 Every real number is either rational or irrational.
The set of real numbers is the set of numbers that are either rational or irrational:
The symbol ℝ is used to represent the set of real numbers. Thus,
Example 5 Recognizing Subsets of the Real Numbers
Consider the following set of numbers:
List the numbers in the set that are
natural numbers.
whole numbers.
integers.
rational numbers.
irrational numbers.
real numbers.
Solution
Natural numbers: The natural numbers are the numbers used for counting. The only natural number in the set is because (9 multiplied by itself, or is 81.)
Whole numbers: The whole numbers consist of the natural numbers and 0. The elements of the set that are whole numbers are 0 and
Integers: The integers consist of the natural numbers, 0, and the negatives of the natural numbers. The elements of the set that are integers are and
Rational numbers: All numbers in the set that can be expressed as the quotient of integers are rational numbers. These include and Furthermore, all numbers in the set that are terminating or repeating decimals are also rational numbers. These include and 7.3.
Irrational numbers: The irrational numbers in the set are and Both and π are only approximately equal to 2.236 and 3.14, respectively. In decimal form, and π neither terminate nor have blocks of repeating digits.
Real numbers: All the numbers in the given set are real numbers.
The real number line is a graph used to represent the set of real numbers. An arbitrary point, called the origin, is labeled 0. Select a point to the right of 0 and label it 1. The distance from 0 to 1 is called the unit distance. Numbers to the right of the origin are positive and numbers to the left of the origin are negative. The real number line is shown in Figure P.5.
Real numbers are graphed on a number line by placing a dot at the correct location for each number. The integers are easiest to locate. In Figure P.6, we’ve graphed six rational numbers and three irrational numbers on a real number line.
Every real number corresponds to a point on the number line and every point on the number line corresponds to a real number. We say that there is a one-to-one correspondence between all the real numbers and all points on a real number line.
Objective 5: Recognize subsets of the real numbers
The Set of Real Numbers
Objective 5Recognize subsets of the real numbers.
Watch Video
The sets that make up the real numbers are summarized in Table P.1. We refer to these sets as subsets of the real numbers, meaning that all elements in each subset are also elements in the set of real numbers.
Notice the use of the symbol ≈ in the examples of irrational numbers. The symbol means “is approximately equal to.” Thus,
We can verify that this is only an approximation by multiplying 1.414214 by itself. The product is very close to, but not exactly, 2:
Not all square roots are irrational. For example, because Thus, is a natural number, a whole number, an integer, and a rational number
The set of real numbers is formed by taking the union of the sets of rational numbers and irrational numbers. Thus, every real number is either rational or irrational, as shown in Figure P.4.
Figure P.4 Every real number is either rational or irrational.
The set of real numbers is the set of numbers that are either rational or irrational:
The symbol ℝ is used to represent the set of real numbers. Thus,
Example 5 Recognizing Subsets of the Real Numbers
Consider the following set of numbers:
List the numbers in the set that are
natural numbers.
whole numbers.
integers.
rational numbers.
irrational numbers.
real numbers.
Solution
Natural numbers: The natural numbers are the numbers used for counting. The only natural number in the set is because (9 multiplied by itself, or is 81.)
Whole numbers: The whole numbers consist of the natural numbers and 0. The elements of the set that are whole numbers are 0 and
Integers: The integers consist of the natural numbers, 0, and the negatives of the natural numbers. The elements of the set that are integers are and
Rational numbers: All numbers in the set that can be expressed as the quotient of integers are rational numbers. These include and Furthermore, all numbers in the set that are terminating or repeating decimals are also rational numbers. These include and 7.3.
Irrational numbers: The irrational numbers in the set are and Both and π are only approximately equal to 2.236 and 3.14, respectively. In decimal form, and π neither terminate nor have blocks of repeating digits.
Real numbers: All the numbers in the given set are real numbers.
The real number line is a graph used to represent the set of real numbers. An arbitrary point, called the origin, is labeled 0. Select a point to the right of 0 and label it 1. The distance from 0 to 1 is called the unit distance. Numbers to the right of the origin are positive and numbers to the left of the origin are negative. The real number line is shown in Figure P.5.
Real numbers are graphed on a number line by placing a dot at the correct location for each number. The integers are easiest to locate. In Figure P.6, we’ve graphed six rational numbers and three irrational numbers on a real number line.
Every real number corresponds to a point on the number line and every point on the number line corresponds to a real number. We say that there is a one-to-one correspondence between all the real numbers and all points on a real number line.
Objective 7: Evaluate absolute value
Absolute Value
Objective 7Evaluate absolute value.
Watch Video
The absolute value of a real number a, denoted by is the distance from 0 to a on the number line. This distance is always taken to be nonnegative. For example, the real number line in Figure P.8 shows that
The absolute value of is 3 because is 3 units from 0 on the number line. The absolute value of 5 is 5 because 5 is 5 units from 0 on the number line. The absolute value of a positive real number or 0 is the number itself. The absolute value of a negative real number, such as is the number without the negative sign.
We can define the absolute value of the real number x without referring to a number line. The algebraic definition of the absolute value of x is given as follows:
Definition of Absolute Value
If x is nonnegative (that is, ), the absolute value of x is the number itself. For example,
Rewrite each expression without absolute value bars:
if
Listed below are several basic properties of absolute value. Each of these properties can be derived from the definition of absolute value.
Properties of Absolute Value
For all real numbers a and b,
(called the triangle inequality).
Objective 7: Evaluate absolute value
Absolute Value
Objective 7Evaluate absolute value.
Watch Video
The absolute value of a real number a, denoted by is the distance from 0 to a on the number line. This distance is always taken to be nonnegative. For example, the real number line in Figure P.8 shows that
The absolute value of is 3 because is 3 units from 0 on the number line. The absolute value of 5 is 5 because 5 is 5 units from 0 on the number line. The absolute value of a positive real number or 0 is the number itself. The absolute value of a negative real number, such as is the number without the negative sign.
We can define the absolute value of the real number x without referring to a number line. The algebraic definition of the absolute value of x is given as follows:
Definition of Absolute Value
If x is nonnegative (that is, ), the absolute value of x is the number itself. For example,
Rewrite each expression without absolute value bars:
if
Listed below are several basic properties of absolute value. Each of these properties can be derived from the definition of absolute value.
Properties of Absolute Value
For all real numbers a and b,
(called the triangle inequality).
Objective 9: Identify properties of the real numbers
Properties of Real Numbers and Algebraic Expressions
Objective 9Identify properties of the real numbers.
Watch Video
When you use your calculator to add two real numbers, you can enter them in any order. The fact that two real numbers can be added in any order is called the commutative property of addition. You probably use this property, as well as other properties of real numbers listed in Table P.2, without giving it much thought. The properties of the real numbers are especially useful when working with algebraic expressions. For each property listed in Table P.2, a, b, and c represent real numbers, variables, or algebraic expressions.
The properties of the real numbers in Table P.2 apply to the operations of addition and multiplication. Subtraction and division are defined in terms of addition and multiplication.
Definitions of Subtraction and Division
Let a and b represent real numbers.
Subtraction: We call the additive inverse or opposite of b.
Division: where We call the multiplicative inverse or reciprocal of b. The quotient of a and b, can be written in the form where a is the numerator and b is the denominator of the fraction.
Because subtraction is defined in terms of adding an inverse, the distributive property can be applied to subtraction:
For example,
Objective 9: Identify properties of the real numbers
Properties of Real Numbers and Algebraic Expressions
Objective 9Identify properties of the real numbers.
Watch Video
When you use your calculator to add two real numbers, you can enter them in any order. The fact that two real numbers can be added in any order is called the commutative property of addition. You probably use this property, as well as other properties of real numbers listed in Table P.2, without giving it much thought. The properties of the real numbers are especially useful when working with algebraic expressions. For each property listed in Table P.2, a, b, and c represent real numbers, variables, or algebraic expressions.
The properties of the real numbers in Table P.2 apply to the operations of addition and multiplication. Subtraction and division are defined in terms of addition and multiplication.
Definitions of Subtraction and Division
Let a and b represent real numbers.
Subtraction: We call the additive inverse or opposite of b.
Division: where We call the multiplicative inverse or reciprocal of b. The quotient of a and b, can be written in the form where a is the numerator and b is the denominator of the fraction.
Because subtraction is defined in terms of adding an inverse, the distributive property can be applied to subtraction:
For example,
Objective 10: Simplify algebraic expressions
Simplifying Algebraic Expressions
Objective 10Simplify algebraic expressions.
Watch Video
The terms of an algebraic expression are those parts that are separated by addition. For example, consider the algebraic expression
which can be expressed as
This expression contains four terms, namely, and
The numerical part of a term is called its coefficient. In the term the 7 is the coefficient. If a term containing one or more variables is written without a coefficient, the coefficient is understood to be 1. Thus, z means If a term is a constant, its coefficient is that constant. Thus, the coefficient of the constant term is
The voice balloons illustrate that negative signs can appear side by side. They can represent the operation of subtraction or the fact that a real number is negative. Here is a list of properties of negatives and how they are applied to algebraic expressions:
Properties of Negatives
Let a and b represent real numbers, variables, or algebraic expressions.
Property
Examples
1.
2.
3.
4.
5.
6.
It is not uncommon to see algebraic expressions with parentheses preceded by a negative sign or subtraction. Properties 5 and 6 in the box, and are related to this situation. An expression of the form can be simplified as follows:
Do you see a fast way to obtain the simplified expression on the right in the preceding equation? If a negative sign or a subtraction symbol appears outside parentheses, drop the parentheses and change the sign of every term within the parentheses. For example,
The terms of an algebraic expression are those parts that are separated by addition. For example, consider the algebraic expression
which can be expressed as
This expression contains four terms, namely, and
The numerical part of a term is called its coefficient. In the term the 7 is the coefficient. If a term containing one or more variables is written without a coefficient, the coefficient is understood to be 1. Thus, z means If a term is a constant, its coefficient is that constant. Thus, the coefficient of the constant term is
The voice balloons illustrate that negative signs can appear side by side. They can represent the operation of subtraction or the fact that a real number is negative. Here is a list of properties of negatives and how they are applied to algebraic expressions:
Properties of Negatives
Let a and b represent real numbers, variables, or algebraic expressions.
Property
Examples
1.
2.
3.
4.
5.
6.
It is not uncommon to see algebraic expressions with parentheses preceded by a negative sign or subtraction. Properties 5 and 6 in the box, and are related to this situation. An expression of the form can be simplified as follows:
Do you see a fast way to obtain the simplified expression on the right in the preceding equation? If a negative sign or a subtraction symbol appears outside parentheses, drop the parentheses and change the sign of every term within the parentheses. For example,
The terms of an algebraic expression are those parts that are separated by addition. For example, consider the algebraic expression
which can be expressed as
This expression contains four terms, namely, and
The numerical part of a term is called its coefficient. In the term the 7 is the coefficient. If a term containing one or more variables is written without a coefficient, the coefficient is understood to be 1. Thus, z means If a term is a constant, its coefficient is that constant. Thus, the coefficient of the constant term is
The voice balloons illustrate that negative signs can appear side by side. They can represent the operation of subtraction or the fact that a real number is negative. Here is a list of properties of negatives and how they are applied to algebraic expressions:
Properties of Negatives
Let a and b represent real numbers, variables, or algebraic expressions.
Property
Examples
1.
2.
3.
4.
5.
6.
It is not uncommon to see algebraic expressions with parentheses preceded by a negative sign or subtraction. Properties 5 and 6 in the box, and are related to this situation. An expression of the form can be simplified as follows:
Do you see a fast way to obtain the simplified expression on the right in the preceding equation? If a negative sign or a subtraction symbol appears outside parentheses, drop the parentheses and change the sign of every term within the parentheses. For example,
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for
3. for and
4. for and
5. for
6. for
7. for
8. for
9. for
10. for
11. for and
12. for and
13. for
14. for
15. for and
16. for and
The formula
expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 17–18, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale.
17.
18.
A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula
describes the ball’s height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 19–20.
19. What was the ball’s height 2 seconds after it was kicked?
20. What was the ball’s height 3 seconds after it was kicked?
In Exercises 21–28, find the intersection of the sets.
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29–34, find the union of the sets.
29.
30.
31.
32.
33.
34.
In Exercises 35–38, list all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
35.
36.
37.
38.
39. Give an example of a whole number that is not a natural number.
40. Give an example of a rational number that is not an integer.
41. Give an example of a number that is an integer, a whole number, and a natural number.
42. Give an example of a number that is a rational number, an integer, and a real number.
Determine whether each statement in Exercises 43–50 is true or false.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–60, rewrite each expression without absolute value bars.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, evaluate each algebraic expression for and
61.
62.
63.
64.
65.
66.
In Exercises 67–74, express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression.
67. 2 and 17
68. 4 and 15
69. and 5
70. and 8
71. and
72. and
73. and
74. and
In Exercises 75–84, state the name of the property illustrated.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
In Exercises 85–96, simplify each algebraic expression.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
In Exercises 97–102, write each algebraic expression without parentheses.
97.
98.
99.
100.
101.
102.
Practice PLUS
In Exercises 103–110, insert either <, >, or in the shaded area to make a true statement.
103.
104.
105.
106.
107.
108.
109.
110.
In Exercises 111–120, use the order of operations to simplify each expression.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number.
121. A number decreased by the sum of the number and four
122. A number decreased by the difference between eight and the number
123. Six times the product of negative five and a number
124. Ten times the product of negative four and a number
125. The difference between the product of five and a number and twice the number
126. The difference between the product of six and a number and negative two times the number
127. The difference between eight times a number and six more than three times the number
128. Eight decreased by three times the sum of a number and six
Application Exercises
The maximum heart rate, in beats per minute, that you should achieve during exercise is 220 minus your age:
The following graph shows the target heart rate ranges for four types of exercise goals. The lower and upper limits of these ranges are fractions of the maximum heart rate, Exercises 129–130 are based on the information in the graph.
129. If your exercise goal is to improve cardiovascular conditioning, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
130. If your exercise goal is to improve overall health, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
The bar graph shows the average cost of tuition and fees at private four-year colleges in the United States.
Here are two formulas that model the data shown in the graph. In each formula, T represents the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000.
Use this information to solve Exercises 131–132.
131.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2018. By how much does each model underestimate or overestimate the actual cost shown for the school year ending in 2018?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2030.
132.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2010. By how much does each underestimate or overestimate the actual cost shown for the school year ending in 2010?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2025.
133. This month you have a total of $6000 in interest-bearing credit card debt, split between a card charging 18% annual interest and a card charging 21% annual interest. If the interest-bearing balance on the card charging 18% is x dollars, then the total interest for the month is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total interest for the month if the balance on the card charging 18% is $4400.
Use the simplified form of the algebraic expression to determine the total interest for the month if the $6000 debt is split evenly between the two cards.
134. It takes you 50 minutes to get to campus. You spend t minutes walking to the bus stop and the rest of the time riding the bus. Your walking rate is 0.06 mile per minute and the bus travels at a rate of 0.5 mile per minute. The total distance walking and traveling by bus is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total distance that you travel if you spend 20 minutes walking to the bus stop.
Use the simplified form of the algebraic expression to determine the total distance you travel if the 50 minutes is split evenly between walking and riding the bus.
135. Read the Blitzer Bonus beginning here. Use the formula
and replace w with your body weight. Using this formula and a calculator, compute your BAC for integers from to . Round to three decimal places. According to this model, how many drinks can you consume in an hour without exceeding the legal measure of drunk driving?
Explaining the Concepts
136. What is an algebraic expression? Give an example with your explanation.
137. If n is a natural number, what does mean? Give an example with your explanation.
138. What does it mean when we say that a formula models real-world phenomena?
139. What is the intersection of sets A and B?
140. What is the union of sets A and B?
141. How do the whole numbers differ from the natural numbers?
142. Can a real number be both rational and irrational? Explain your answer.
143. If you are given two real numbers, explain how to determine which is the lesser.
Critical Thinking Exercises
Make Sense? In Exercises 144–147, determine whether each statement makes sense or does not make sense, and explain your reasoning.
144. My mathematical model describes the data for tuition and fees at public four-year colleges for the past 20 years extremely well, so it will serve as an accurate prediction for the cost of public colleges in 2050.
145. A model that describes the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000 cannot be used to estimate the cost of private education for the school year ending in 2000.
146. Regardless of what real numbers I substitute for x and y, I will always obtain zero when evaluating
147. Just as the commutative properties change groupings, the associative properties change order.
In Exercises 148–155, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
148. Every rational number is an integer.
149. Some whole numbers are not integers.
150. Some rational numbers are not positive.
151. Irrational numbers cannot be negative.
152. The term x has no coefficient.
153.
154.
155.
In Exercises 156–158, insert either < or > in the shaded area between the numbers to make the statement true.
156.
157.
158.
Preview Exercises
Exercises 159–161 will help you prepare for the material covered in the next section.
159. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base?
160. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?
161. If 6.2 is multiplied by what does this multiplication do to the decimal point in 6.2?
P.1: Exercise Set
P.1 Exercise Set
Practice Exercises
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for
3. for and
4. for and
5. for
6. for
7. for
8. for
9. for
10. for
11. for and
12. for and
13. for
14. for
15. for and
16. for and
The formula
expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 17–18, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale.
17.
18.
A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula
describes the ball’s height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 19–20.
19. What was the ball’s height 2 seconds after it was kicked?
20. What was the ball’s height 3 seconds after it was kicked?
In Exercises 21–28, find the intersection of the sets.
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29–34, find the union of the sets.
29.
30.
31.
32.
33.
34.
In Exercises 35–38, list all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
35.
36.
37.
38.
39. Give an example of a whole number that is not a natural number.
40. Give an example of a rational number that is not an integer.
41. Give an example of a number that is an integer, a whole number, and a natural number.
42. Give an example of a number that is a rational number, an integer, and a real number.
Determine whether each statement in Exercises 43–50 is true or false.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–60, rewrite each expression without absolute value bars.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, evaluate each algebraic expression for and
61.
62.
63.
64.
65.
66.
In Exercises 67–74, express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression.
67. 2 and 17
68. 4 and 15
69. and 5
70. and 8
71. and
72. and
73. and
74. and
In Exercises 75–84, state the name of the property illustrated.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
In Exercises 85–96, simplify each algebraic expression.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
In Exercises 97–102, write each algebraic expression without parentheses.
97.
98.
99.
100.
101.
102.
Practice PLUS
In Exercises 103–110, insert either <, >, or in the shaded area to make a true statement.
103.
104.
105.
106.
107.
108.
109.
110.
In Exercises 111–120, use the order of operations to simplify each expression.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number.
121. A number decreased by the sum of the number and four
122. A number decreased by the difference between eight and the number
123. Six times the product of negative five and a number
124. Ten times the product of negative four and a number
125. The difference between the product of five and a number and twice the number
126. The difference between the product of six and a number and negative two times the number
127. The difference between eight times a number and six more than three times the number
128. Eight decreased by three times the sum of a number and six
Application Exercises
The maximum heart rate, in beats per minute, that you should achieve during exercise is 220 minus your age:
The following graph shows the target heart rate ranges for four types of exercise goals. The lower and upper limits of these ranges are fractions of the maximum heart rate, Exercises 129–130 are based on the information in the graph.
129. If your exercise goal is to improve cardiovascular conditioning, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
130. If your exercise goal is to improve overall health, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
The bar graph shows the average cost of tuition and fees at private four-year colleges in the United States.
Here are two formulas that model the data shown in the graph. In each formula, T represents the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000.
Use this information to solve Exercises 131–132.
131.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2018. By how much does each model underestimate or overestimate the actual cost shown for the school year ending in 2018?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2030.
132.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2010. By how much does each underestimate or overestimate the actual cost shown for the school year ending in 2010?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2025.
133. This month you have a total of $6000 in interest-bearing credit card debt, split between a card charging 18% annual interest and a card charging 21% annual interest. If the interest-bearing balance on the card charging 18% is x dollars, then the total interest for the month is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total interest for the month if the balance on the card charging 18% is $4400.
Use the simplified form of the algebraic expression to determine the total interest for the month if the $6000 debt is split evenly between the two cards.
134. It takes you 50 minutes to get to campus. You spend t minutes walking to the bus stop and the rest of the time riding the bus. Your walking rate is 0.06 mile per minute and the bus travels at a rate of 0.5 mile per minute. The total distance walking and traveling by bus is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total distance that you travel if you spend 20 minutes walking to the bus stop.
Use the simplified form of the algebraic expression to determine the total distance you travel if the 50 minutes is split evenly between walking and riding the bus.
135. Read the Blitzer Bonus beginning here. Use the formula
and replace w with your body weight. Using this formula and a calculator, compute your BAC for integers from to . Round to three decimal places. According to this model, how many drinks can you consume in an hour without exceeding the legal measure of drunk driving?
Explaining the Concepts
136. What is an algebraic expression? Give an example with your explanation.
137. If n is a natural number, what does mean? Give an example with your explanation.
138. What does it mean when we say that a formula models real-world phenomena?
139. What is the intersection of sets A and B?
140. What is the union of sets A and B?
141. How do the whole numbers differ from the natural numbers?
142. Can a real number be both rational and irrational? Explain your answer.
143. If you are given two real numbers, explain how to determine which is the lesser.
Critical Thinking Exercises
Make Sense? In Exercises 144–147, determine whether each statement makes sense or does not make sense, and explain your reasoning.
144. My mathematical model describes the data for tuition and fees at public four-year colleges for the past 20 years extremely well, so it will serve as an accurate prediction for the cost of public colleges in 2050.
145. A model that describes the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000 cannot be used to estimate the cost of private education for the school year ending in 2000.
146. Regardless of what real numbers I substitute for x and y, I will always obtain zero when evaluating
147. Just as the commutative properties change groupings, the associative properties change order.
In Exercises 148–155, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
148. Every rational number is an integer.
149. Some whole numbers are not integers.
150. Some rational numbers are not positive.
151. Irrational numbers cannot be negative.
152. The term x has no coefficient.
153.
154.
155.
In Exercises 156–158, insert either < or > in the shaded area between the numbers to make the statement true.
156.
157.
158.
Preview Exercises
Exercises 159–161 will help you prepare for the material covered in the next section.
159. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base?
160. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?
161. If 6.2 is multiplied by what does this multiplication do to the decimal point in 6.2?
P.1: Exercise Set
P.1 Exercise Set
Practice Exercises
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for
3. for and
4. for and
5. for
6. for
7. for
8. for
9. for
10. for
11. for and
12. for and
13. for
14. for
15. for and
16. for and
The formula
expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 17–18, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale.
17.
18.
A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula
describes the ball’s height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 19–20.
19. What was the ball’s height 2 seconds after it was kicked?
20. What was the ball’s height 3 seconds after it was kicked?
In Exercises 21–28, find the intersection of the sets.
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29–34, find the union of the sets.
29.
30.
31.
32.
33.
34.
In Exercises 35–38, list all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
35.
36.
37.
38.
39. Give an example of a whole number that is not a natural number.
40. Give an example of a rational number that is not an integer.
41. Give an example of a number that is an integer, a whole number, and a natural number.
42. Give an example of a number that is a rational number, an integer, and a real number.
Determine whether each statement in Exercises 43–50 is true or false.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–60, rewrite each expression without absolute value bars.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, evaluate each algebraic expression for and
61.
62.
63.
64.
65.
66.
In Exercises 67–74, express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression.
67. 2 and 17
68. 4 and 15
69. and 5
70. and 8
71. and
72. and
73. and
74. and
In Exercises 75–84, state the name of the property illustrated.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
In Exercises 85–96, simplify each algebraic expression.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
In Exercises 97–102, write each algebraic expression without parentheses.
97.
98.
99.
100.
101.
102.
Practice PLUS
In Exercises 103–110, insert either <, >, or in the shaded area to make a true statement.
103.
104.
105.
106.
107.
108.
109.
110.
In Exercises 111–120, use the order of operations to simplify each expression.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number.
121. A number decreased by the sum of the number and four
122. A number decreased by the difference between eight and the number
123. Six times the product of negative five and a number
124. Ten times the product of negative four and a number
125. The difference between the product of five and a number and twice the number
126. The difference between the product of six and a number and negative two times the number
127. The difference between eight times a number and six more than three times the number
128. Eight decreased by three times the sum of a number and six
Application Exercises
The maximum heart rate, in beats per minute, that you should achieve during exercise is 220 minus your age:
The following graph shows the target heart rate ranges for four types of exercise goals. The lower and upper limits of these ranges are fractions of the maximum heart rate, Exercises 129–130 are based on the information in the graph.
129. If your exercise goal is to improve cardiovascular conditioning, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
130. If your exercise goal is to improve overall health, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
The bar graph shows the average cost of tuition and fees at private four-year colleges in the United States.
Here are two formulas that model the data shown in the graph. In each formula, T represents the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000.
Use this information to solve Exercises 131–132.
131.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2018. By how much does each model underestimate or overestimate the actual cost shown for the school year ending in 2018?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2030.
132.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2010. By how much does each underestimate or overestimate the actual cost shown for the school year ending in 2010?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2025.
133. This month you have a total of $6000 in interest-bearing credit card debt, split between a card charging 18% annual interest and a card charging 21% annual interest. If the interest-bearing balance on the card charging 18% is x dollars, then the total interest for the month is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total interest for the month if the balance on the card charging 18% is $4400.
Use the simplified form of the algebraic expression to determine the total interest for the month if the $6000 debt is split evenly between the two cards.
134. It takes you 50 minutes to get to campus. You spend t minutes walking to the bus stop and the rest of the time riding the bus. Your walking rate is 0.06 mile per minute and the bus travels at a rate of 0.5 mile per minute. The total distance walking and traveling by bus is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total distance that you travel if you spend 20 minutes walking to the bus stop.
Use the simplified form of the algebraic expression to determine the total distance you travel if the 50 minutes is split evenly between walking and riding the bus.
135. Read the Blitzer Bonus beginning here. Use the formula
and replace w with your body weight. Using this formula and a calculator, compute your BAC for integers from to . Round to three decimal places. According to this model, how many drinks can you consume in an hour without exceeding the legal measure of drunk driving?
Explaining the Concepts
136. What is an algebraic expression? Give an example with your explanation.
137. If n is a natural number, what does mean? Give an example with your explanation.
138. What does it mean when we say that a formula models real-world phenomena?
139. What is the intersection of sets A and B?
140. What is the union of sets A and B?
141. How do the whole numbers differ from the natural numbers?
142. Can a real number be both rational and irrational? Explain your answer.
143. If you are given two real numbers, explain how to determine which is the lesser.
Critical Thinking Exercises
Make Sense? In Exercises 144–147, determine whether each statement makes sense or does not make sense, and explain your reasoning.
144. My mathematical model describes the data for tuition and fees at public four-year colleges for the past 20 years extremely well, so it will serve as an accurate prediction for the cost of public colleges in 2050.
145. A model that describes the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000 cannot be used to estimate the cost of private education for the school year ending in 2000.
146. Regardless of what real numbers I substitute for x and y, I will always obtain zero when evaluating
147. Just as the commutative properties change groupings, the associative properties change order.
In Exercises 148–155, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
148. Every rational number is an integer.
149. Some whole numbers are not integers.
150. Some rational numbers are not positive.
151. Irrational numbers cannot be negative.
152. The term x has no coefficient.
153.
154.
155.
In Exercises 156–158, insert either < or > in the shaded area between the numbers to make the statement true.
156.
157.
158.
Preview Exercises
Exercises 159–161 will help you prepare for the material covered in the next section.
159. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base?
160. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?
161. If 6.2 is multiplied by what does this multiplication do to the decimal point in 6.2?
P.1: Exercise Set
P.1 Exercise Set
Practice Exercises
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for
3. for and
4. for and
5. for
6. for
7. for
8. for
9. for
10. for
11. for and
12. for and
13. for
14. for
15. for and
16. for and
The formula
expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 17–18, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale.
17.
18.
A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula
describes the ball’s height above the ground, h, in feet, t seconds after it was kicked. Use this formula to solve Exercises 19–20.
19. What was the ball’s height 2 seconds after it was kicked?
20. What was the ball’s height 3 seconds after it was kicked?
In Exercises 21–28, find the intersection of the sets.
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29–34, find the union of the sets.
29.
30.
31.
32.
33.
34.
In Exercises 35–38, list all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
35.
36.
37.
38.
39. Give an example of a whole number that is not a natural number.
40. Give an example of a rational number that is not an integer.
41. Give an example of a number that is an integer, a whole number, and a natural number.
42. Give an example of a number that is a rational number, an integer, and a real number.
Determine whether each statement in Exercises 43–50 is true or false.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–60, rewrite each expression without absolute value bars.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, evaluate each algebraic expression for and
61.
62.
63.
64.
65.
66.
In Exercises 67–74, express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression.
67. 2 and 17
68. 4 and 15
69. and 5
70. and 8
71. and
72. and
73. and
74. and
In Exercises 75–84, state the name of the property illustrated.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
In Exercises 85–96, simplify each algebraic expression.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
In Exercises 97–102, write each algebraic expression without parentheses.
97.
98.
99.
100.
101.
102.
Practice PLUS
In Exercises 103–110, insert either <, >, or in the shaded area to make a true statement.
103.
104.
105.
106.
107.
108.
109.
110.
In Exercises 111–120, use the order of operations to simplify each expression.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number.
121. A number decreased by the sum of the number and four
122. A number decreased by the difference between eight and the number
123. Six times the product of negative five and a number
124. Ten times the product of negative four and a number
125. The difference between the product of five and a number and twice the number
126. The difference between the product of six and a number and negative two times the number
127. The difference between eight times a number and six more than three times the number
128. Eight decreased by three times the sum of a number and six
Application Exercises
The maximum heart rate, in beats per minute, that you should achieve during exercise is 220 minus your age:
The following graph shows the target heart rate ranges for four types of exercise goals. The lower and upper limits of these ranges are fractions of the maximum heart rate, Exercises 129–130 are based on the information in the graph.
129. If your exercise goal is to improve cardiovascular conditioning, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 20-year-old with this exercise goal?
130. If your exercise goal is to improve overall health, the graph shows the following range for target heart rate, H, in beats per minute:
What is the lower limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
What is the upper limit of the heart rate range, in beats per minute, for a 30-year-old with this exercise goal?
The bar graph shows the average cost of tuition and fees at private four-year colleges in the United States.
Here are two formulas that model the data shown in the graph. In each formula, T represents the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000.
Use this information to solve Exercises 131–132.
131.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2018. By how much does each model underestimate or overestimate the actual cost shown for the school year ending in 2018?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2030.
132.
Use each formula to find the average cost of tuition and fees at private U.S. colleges for the school year ending in 2010. By how much does each underestimate or overestimate the actual cost shown for the school year ending in 2010?
Use model 2 to project the average cost of tuition and fees at private U.S. colleges for the school year ending in 2025.
133. This month you have a total of $6000 in interest-bearing credit card debt, split between a card charging 18% annual interest and a card charging 21% annual interest. If the interest-bearing balance on the card charging 18% is x dollars, then the total interest for the month is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total interest for the month if the balance on the card charging 18% is $4400.
Use the simplified form of the algebraic expression to determine the total interest for the month if the $6000 debt is split evenly between the two cards.
134. It takes you 50 minutes to get to campus. You spend t minutes walking to the bus stop and the rest of the time riding the bus. Your walking rate is 0.06 mile per minute and the bus travels at a rate of 0.5 mile per minute. The total distance walking and traveling by bus is given by the algebraic expression
Simplify the algebraic expression.
Use each form of the algebraic expression to determine the total distance that you travel if you spend 20 minutes walking to the bus stop.
Use the simplified form of the algebraic expression to determine the total distance you travel if the 50 minutes is split evenly between walking and riding the bus.
135. Read the Blitzer Bonus beginning here. Use the formula
and replace w with your body weight. Using this formula and a calculator, compute your BAC for integers from to . Round to three decimal places. According to this model, how many drinks can you consume in an hour without exceeding the legal measure of drunk driving?
Explaining the Concepts
136. What is an algebraic expression? Give an example with your explanation.
137. If n is a natural number, what does mean? Give an example with your explanation.
138. What does it mean when we say that a formula models real-world phenomena?
139. What is the intersection of sets A and B?
140. What is the union of sets A and B?
141. How do the whole numbers differ from the natural numbers?
142. Can a real number be both rational and irrational? Explain your answer.
143. If you are given two real numbers, explain how to determine which is the lesser.
Critical Thinking Exercises
Make Sense? In Exercises 144–147, determine whether each statement makes sense or does not make sense, and explain your reasoning.
144. My mathematical model describes the data for tuition and fees at public four-year colleges for the past 20 years extremely well, so it will serve as an accurate prediction for the cost of public colleges in 2050.
145. A model that describes the average cost of tuition and fees at private U.S. colleges for the school year ending x years after 2000 cannot be used to estimate the cost of private education for the school year ending in 2000.
146. Regardless of what real numbers I substitute for x and y, I will always obtain zero when evaluating
147. Just as the commutative properties change groupings, the associative properties change order.
In Exercises 148–155, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
148. Every rational number is an integer.
149. Some whole numbers are not integers.
150. Some rational numbers are not positive.
151. Irrational numbers cannot be negative.
152. The term x has no coefficient.
153.
154.
155.
In Exercises 156–158, insert either < or > in the shaded area between the numbers to make the statement true.
156.
157.
158.
Preview Exercises
Exercises 159–161 will help you prepare for the material covered in the next section.
159. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when multiplying exponential expressions with the same base?
160. In parts (a) and (b), complete each statement.
Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?
161. If 6.2 is multiplied by what does this multiplication do to the decimal point in 6.2?
Objective 1: Use properties of exponents
Properties of Exponents
Objective 1Use properties of exponents.
The major properties of exponents are summarized in the box that follows.
Properties of Exponents
Property
Examples
The Negative-Exponent Rule
If b is any real number other than 0 and n is a natural number, then
The Zero-Exponent Rule
If b is any real number other than 0,
The Product Rule
If b is a real number or algebraic expression, and m and n are integers,
When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.
The Power Rule
If b is a real number or algebraic expression, and m and n are integers,
When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.
The Quotient Rule
If b is a nonzero real number or algebraic expression, and m and n are integers,
When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.
Products Raised to Powers
If a and b are real numbers or algebraic expressions, and n is an integer,
When a product is raised to a power, raise each factor to that power.
Quotients Raised to Powers
If a and b are real numbers, or algebraic expressions, and n is an integer,
When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.
Objective 1: Use properties of exponents
Properties of Exponents
Objective 1Use properties of exponents.
The major properties of exponents are summarized in the box that follows.
Properties of Exponents
Property
Examples
The Negative-Exponent Rule
If b is any real number other than 0 and n is a natural number, then
The Zero-Exponent Rule
If b is any real number other than 0,
The Product Rule
If b is a real number or algebraic expression, and m and n are integers,
When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.
The Power Rule
If b is a real number or algebraic expression, and m and n are integers,
When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.
The Quotient Rule
If b is a nonzero real number or algebraic expression, and m and n are integers,
When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.
Products Raised to Powers
If a and b are real numbers or algebraic expressions, and n is an integer,
When a product is raised to a power, raise each factor to that power.
Quotients Raised to Powers
If a and b are real numbers, or algebraic expressions, and n is an integer,
When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.
Objective 2: Simplify exponential expressions
Simplifying Exponential Expressions
Objective 2Simplify exponential expressions.
Watch Video
Properties of exponents are used to simplify exponential expressions. An exponential expression is simplified when
No parentheses appear.
No powers are raised to powers.
Each base occurs only once.
No negative or zero exponents appear.
Simplifying Exponential Expressions
Example
1. If necessary, remove parentheses by using
2. If necessary, simplify powers to powers by using
3. If necessary, be sure that each base appears only once by using
4. If necessary, rewrite exponential expressions with zero powers as Furthermore, write the answer with positive exponents by using
The following example shows how to simplify exponential expressions. Throughout the example, assume that no variable in a denominator is equal to zero.
Properties of exponents are used to simplify exponential expressions. An exponential expression is simplified when
No parentheses appear.
No powers are raised to powers.
Each base occurs only once.
No negative or zero exponents appear.
Simplifying Exponential Expressions
Example
1. If necessary, remove parentheses by using
2. If necessary, simplify powers to powers by using
3. If necessary, be sure that each base appears only once by using
4. If necessary, rewrite exponential expressions with zero powers as Furthermore, write the answer with positive exponents by using
The following example shows how to simplify exponential expressions. Throughout the example, assume that no variable in a denominator is equal to zero.
Earth is a 4.5-billion-year-old ball of rock orbiting the Sun. Because a billion is (see Table P.3), the age of our world, in years, can be expressed as
The number is written in a form called scientific notation.
Table P.3 Names of Large Numbers
hundred
thousand
million
billion
trillion
quadrillion
quintillion
sextillion
septillion
octillion
nonillion
googol
googolplex
Scientific Notation
A number is written in scientific notation when it is expressed in the form
where the absolute value of a is greater than or equal to 1 and less than 10 and n is an integer.
It is customary to use the multiplication symbol, ×, rather than a dot, when writing a number in scientific notation.
Converting from Scientific to Decimal Notation
Here are two examples of numbers in scientific notation:
Do you see that the number with the positive exponent is relatively large and the number with the negative exponent is relatively small?
We can use n, the exponent on the 10 in to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the rightn places. If n is negative, move the decimal point in a to the left places.
Example 2 Converting from Scientific to Decimal Notation
Write each number in decimal notation:
Solution
In each case, we use the exponent on the 10 to determine how far to move the decimal point and in which direction. In parts (a) and (b), the exponent is positive, so we move the decimal point to the right. In parts (c) and (d), the exponent is negative, so we move the decimal point to the left.
To convert from decimal notation to scientific notation, we reverse the procedure of Example 2.
Converting from Decimal to Scientific Notation
Write the number in the form
Determine a, the numerical factor. Move the decimal point in the given number to obtain a number whose absolute value is between 1 and 10, including 1.
Determine n, the exponent on The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the decimal point was moved to the left, negative if the decimal point was moved to the right, and 0 if the decimal point was not moved.
Example 3 Converting from Decimal Notation to Scientific Notation
Perform the indicated computations, writing the answers in scientific notation:
Applications: Putting Numbers in Perspective
Due to tax cuts and spending increases, the United States began accumulating large deficits in the 1980s. To finance the deficit, the government had borrowed $25.5 trillion as of May 2020. The graph in Figure P.10 shows the national debt increasing over time. The amount shown for 2020 is midyear; the big increase in the national debt in the early part of the year was due in part to the economic impact of the COVID-19 pandemic.
Example 6 shows how we can use scientific notation to comprehend the meaning of a number such as 25.5 trillion.
Example 6 The National Debt
As of May 2020, the national debt was $25.5 trillion, or dollars. At that time, the U.S. population was approximately 331,000,000 (331 million), or If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay?
Solution
The amount each citizen must pay is the total debt, dollars, divided by the number of citizens,
Every U.S. citizen would have to pay approximately $77,000 to the federal government to pay off the national debt. At the end of 2019, the median yearly income of full-time workers in the United States was $48,672 (Source: Bureau of Labor Statistics); a worker with this income would have to pay more than 1.5 years’ salary to pay off the national debt.
At the end of 2010, the national debt was $13.5 trillion. At that time, the U.S. population was approximately 309 million. Write 13.5 trillion and 309 million in scientific notation. Then determine how much each citizen would have had to pay if the national debt in 2010 had been evenly divided among every individual in the 2010 population. Round to the nearest thousand dollars. Compare this amount to the $77,000 per person in 2020 from Example 6. How much did each individual’s share of the national debt increase between 2010 and 2020?
An Application: Black Holes in Space
The concept of a black hole, a region in space where matter appears to vanish, intrigues scientists and nonscientists alike. Scientists theorize that when massive stars run out of nuclear fuel, they begin to collapse under the force of their own gravity. As the star collapses, its density increases. In turn, the force of gravity increases so tremendously that even light cannot escape from the star. Consequently, it appears black.
A mathematical model, called the Schwarzchild formula, describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. This model forms the basis of our next example.
Example 7 An Application of Scientific Notation
Use the Schwarzchild formula
where
to determine to what length the radius of the Sun must be reduced for it to become a black hole. The Sun’s mass is approximately
Solution
Although the Sun is not massive enough to become a black hole (its radius is approximately 700,000 kilometers), the Schwarzchild model theoretically indicates that if the Sun’s radius were reduced to approximately 2978 meters, that is, about its present size, it would become a black hole.
The speed of blood, S, in centimeters per second, located r centimeters from the central axis of an artery is modeled by
Find the speed of blood at the central axis of this artery.
Objective 3: Use scientific notation
Scientific Notation
Objective 3Use scientific notation.
Watch Video
Earth is a 4.5-billion-year-old ball of rock orbiting the Sun. Because a billion is (see Table P.3), the age of our world, in years, can be expressed as
The number is written in a form called scientific notation.
Table P.3 Names of Large Numbers
hundred
thousand
million
billion
trillion
quadrillion
quintillion
sextillion
septillion
octillion
nonillion
googol
googolplex
Scientific Notation
A number is written in scientific notation when it is expressed in the form
where the absolute value of a is greater than or equal to 1 and less than 10 and n is an integer.
It is customary to use the multiplication symbol, ×, rather than a dot, when writing a number in scientific notation.
Converting from Scientific to Decimal Notation
Here are two examples of numbers in scientific notation:
Do you see that the number with the positive exponent is relatively large and the number with the negative exponent is relatively small?
We can use n, the exponent on the 10 in to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the rightn places. If n is negative, move the decimal point in a to the left places.
Example 2 Converting from Scientific to Decimal Notation
Write each number in decimal notation:
Solution
In each case, we use the exponent on the 10 to determine how far to move the decimal point and in which direction. In parts (a) and (b), the exponent is positive, so we move the decimal point to the right. In parts (c) and (d), the exponent is negative, so we move the decimal point to the left.
To convert from decimal notation to scientific notation, we reverse the procedure of Example 2.
Converting from Decimal to Scientific Notation
Write the number in the form
Determine a, the numerical factor. Move the decimal point in the given number to obtain a number whose absolute value is between 1 and 10, including 1.
Determine n, the exponent on The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the decimal point was moved to the left, negative if the decimal point was moved to the right, and 0 if the decimal point was not moved.
Example 3 Converting from Decimal Notation to Scientific Notation
Perform the indicated computations, writing the answers in scientific notation:
Applications: Putting Numbers in Perspective
Due to tax cuts and spending increases, the United States began accumulating large deficits in the 1980s. To finance the deficit, the government had borrowed $25.5 trillion as of May 2020. The graph in Figure P.10 shows the national debt increasing over time. The amount shown for 2020 is midyear; the big increase in the national debt in the early part of the year was due in part to the economic impact of the COVID-19 pandemic.
Example 6 shows how we can use scientific notation to comprehend the meaning of a number such as 25.5 trillion.
Example 6 The National Debt
As of May 2020, the national debt was $25.5 trillion, or dollars. At that time, the U.S. population was approximately 331,000,000 (331 million), or If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay?
Solution
The amount each citizen must pay is the total debt, dollars, divided by the number of citizens,
Every U.S. citizen would have to pay approximately $77,000 to the federal government to pay off the national debt. At the end of 2019, the median yearly income of full-time workers in the United States was $48,672 (Source: Bureau of Labor Statistics); a worker with this income would have to pay more than 1.5 years’ salary to pay off the national debt.
At the end of 2010, the national debt was $13.5 trillion. At that time, the U.S. population was approximately 309 million. Write 13.5 trillion and 309 million in scientific notation. Then determine how much each citizen would have had to pay if the national debt in 2010 had been evenly divided among every individual in the 2010 population. Round to the nearest thousand dollars. Compare this amount to the $77,000 per person in 2020 from Example 6. How much did each individual’s share of the national debt increase between 2010 and 2020?
An Application: Black Holes in Space
The concept of a black hole, a region in space where matter appears to vanish, intrigues scientists and nonscientists alike. Scientists theorize that when massive stars run out of nuclear fuel, they begin to collapse under the force of their own gravity. As the star collapses, its density increases. In turn, the force of gravity increases so tremendously that even light cannot escape from the star. Consequently, it appears black.
A mathematical model, called the Schwarzchild formula, describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. This model forms the basis of our next example.
Example 7 An Application of Scientific Notation
Use the Schwarzchild formula
where
to determine to what length the radius of the Sun must be reduced for it to become a black hole. The Sun’s mass is approximately
Solution
Although the Sun is not massive enough to become a black hole (its radius is approximately 700,000 kilometers), the Schwarzchild model theoretically indicates that if the Sun’s radius were reduced to approximately 2978 meters, that is, about its present size, it would become a black hole.
The speed of blood, S, in centimeters per second, located r centimeters from the central axis of an artery is modeled by
Find the speed of blood at the central axis of this artery.
Objective 3: Use scientific notation
Scientific Notation
Objective 3Use scientific notation.
Watch Video
Earth is a 4.5-billion-year-old ball of rock orbiting the Sun. Because a billion is (see Table P.3), the age of our world, in years, can be expressed as
The number is written in a form called scientific notation.
Table P.3 Names of Large Numbers
hundred
thousand
million
billion
trillion
quadrillion
quintillion
sextillion
septillion
octillion
nonillion
googol
googolplex
Scientific Notation
A number is written in scientific notation when it is expressed in the form
where the absolute value of a is greater than or equal to 1 and less than 10 and n is an integer.
It is customary to use the multiplication symbol, ×, rather than a dot, when writing a number in scientific notation.
Converting from Scientific to Decimal Notation
Here are two examples of numbers in scientific notation:
Do you see that the number with the positive exponent is relatively large and the number with the negative exponent is relatively small?
We can use n, the exponent on the 10 in to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the rightn places. If n is negative, move the decimal point in a to the left places.
Example 2 Converting from Scientific to Decimal Notation
Write each number in decimal notation:
Solution
In each case, we use the exponent on the 10 to determine how far to move the decimal point and in which direction. In parts (a) and (b), the exponent is positive, so we move the decimal point to the right. In parts (c) and (d), the exponent is negative, so we move the decimal point to the left.
To convert from decimal notation to scientific notation, we reverse the procedure of Example 2.
Converting from Decimal to Scientific Notation
Write the number in the form
Determine a, the numerical factor. Move the decimal point in the given number to obtain a number whose absolute value is between 1 and 10, including 1.
Determine n, the exponent on The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the decimal point was moved to the left, negative if the decimal point was moved to the right, and 0 if the decimal point was not moved.
Example 3 Converting from Decimal Notation to Scientific Notation
Perform the indicated computations, writing the answers in scientific notation:
Applications: Putting Numbers in Perspective
Due to tax cuts and spending increases, the United States began accumulating large deficits in the 1980s. To finance the deficit, the government had borrowed $25.5 trillion as of May 2020. The graph in Figure P.10 shows the national debt increasing over time. The amount shown for 2020 is midyear; the big increase in the national debt in the early part of the year was due in part to the economic impact of the COVID-19 pandemic.
Example 6 shows how we can use scientific notation to comprehend the meaning of a number such as 25.5 trillion.
Example 6 The National Debt
As of May 2020, the national debt was $25.5 trillion, or dollars. At that time, the U.S. population was approximately 331,000,000 (331 million), or If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay?
Solution
The amount each citizen must pay is the total debt, dollars, divided by the number of citizens,
Every U.S. citizen would have to pay approximately $77,000 to the federal government to pay off the national debt. At the end of 2019, the median yearly income of full-time workers in the United States was $48,672 (Source: Bureau of Labor Statistics); a worker with this income would have to pay more than 1.5 years’ salary to pay off the national debt.
At the end of 2010, the national debt was $13.5 trillion. At that time, the U.S. population was approximately 309 million. Write 13.5 trillion and 309 million in scientific notation. Then determine how much each citizen would have had to pay if the national debt in 2010 had been evenly divided among every individual in the 2010 population. Round to the nearest thousand dollars. Compare this amount to the $77,000 per person in 2020 from Example 6. How much did each individual’s share of the national debt increase between 2010 and 2020?
An Application: Black Holes in Space
The concept of a black hole, a region in space where matter appears to vanish, intrigues scientists and nonscientists alike. Scientists theorize that when massive stars run out of nuclear fuel, they begin to collapse under the force of their own gravity. As the star collapses, its density increases. In turn, the force of gravity increases so tremendously that even light cannot escape from the star. Consequently, it appears black.
A mathematical model, called the Schwarzchild formula, describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. This model forms the basis of our next example.
Example 7 An Application of Scientific Notation
Use the Schwarzchild formula
where
to determine to what length the radius of the Sun must be reduced for it to become a black hole. The Sun’s mass is approximately
Solution
Although the Sun is not massive enough to become a black hole (its radius is approximately 700,000 kilometers), the Schwarzchild model theoretically indicates that if the Sun’s radius were reduced to approximately 2978 meters, that is, about its present size, it would become a black hole.
The speed of blood, S, in centimeters per second, located r centimeters from the central axis of an artery is modeled by
Find the speed of blood at the central axis of this artery.
Objective 3: Use scientific notation
Scientific Notation
Objective 3Use scientific notation.
Watch Video
Earth is a 4.5-billion-year-old ball of rock orbiting the Sun. Because a billion is (see Table P.3), the age of our world, in years, can be expressed as
The number is written in a form called scientific notation.
Table P.3 Names of Large Numbers
hundred
thousand
million
billion
trillion
quadrillion
quintillion
sextillion
septillion
octillion
nonillion
googol
googolplex
Scientific Notation
A number is written in scientific notation when it is expressed in the form
where the absolute value of a is greater than or equal to 1 and less than 10 and n is an integer.
It is customary to use the multiplication symbol, ×, rather than a dot, when writing a number in scientific notation.
Converting from Scientific to Decimal Notation
Here are two examples of numbers in scientific notation:
Do you see that the number with the positive exponent is relatively large and the number with the negative exponent is relatively small?
We can use n, the exponent on the 10 in to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the rightn places. If n is negative, move the decimal point in a to the left places.
Example 2 Converting from Scientific to Decimal Notation
Write each number in decimal notation:
Solution
In each case, we use the exponent on the 10 to determine how far to move the decimal point and in which direction. In parts (a) and (b), the exponent is positive, so we move the decimal point to the right. In parts (c) and (d), the exponent is negative, so we move the decimal point to the left.
To convert from decimal notation to scientific notation, we reverse the procedure of Example 2.
Converting from Decimal to Scientific Notation
Write the number in the form
Determine a, the numerical factor. Move the decimal point in the given number to obtain a number whose absolute value is between 1 and 10, including 1.
Determine n, the exponent on The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the decimal point was moved to the left, negative if the decimal point was moved to the right, and 0 if the decimal point was not moved.
Example 3 Converting from Decimal Notation to Scientific Notation
Perform the indicated computations, writing the answers in scientific notation:
Applications: Putting Numbers in Perspective
Due to tax cuts and spending increases, the United States began accumulating large deficits in the 1980s. To finance the deficit, the government had borrowed $25.5 trillion as of May 2020. The graph in Figure P.10 shows the national debt increasing over time. The amount shown for 2020 is midyear; the big increase in the national debt in the early part of the year was due in part to the economic impact of the COVID-19 pandemic.
Example 6 shows how we can use scientific notation to comprehend the meaning of a number such as 25.5 trillion.
Example 6 The National Debt
As of May 2020, the national debt was $25.5 trillion, or dollars. At that time, the U.S. population was approximately 331,000,000 (331 million), or If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay?
Solution
The amount each citizen must pay is the total debt, dollars, divided by the number of citizens,
Every U.S. citizen would have to pay approximately $77,000 to the federal government to pay off the national debt. At the end of 2019, the median yearly income of full-time workers in the United States was $48,672 (Source: Bureau of Labor Statistics); a worker with this income would have to pay more than 1.5 years’ salary to pay off the national debt.
At the end of 2010, the national debt was $13.5 trillion. At that time, the U.S. population was approximately 309 million. Write 13.5 trillion and 309 million in scientific notation. Then determine how much each citizen would have had to pay if the national debt in 2010 had been evenly divided among every individual in the 2010 population. Round to the nearest thousand dollars. Compare this amount to the $77,000 per person in 2020 from Example 6. How much did each individual’s share of the national debt increase between 2010 and 2020?
An Application: Black Holes in Space
The concept of a black hole, a region in space where matter appears to vanish, intrigues scientists and nonscientists alike. Scientists theorize that when massive stars run out of nuclear fuel, they begin to collapse under the force of their own gravity. As the star collapses, its density increases. In turn, the force of gravity increases so tremendously that even light cannot escape from the star. Consequently, it appears black.
A mathematical model, called the Schwarzchild formula, describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. This model forms the basis of our next example.
Example 7 An Application of Scientific Notation
Use the Schwarzchild formula
where
to determine to what length the radius of the Sun must be reduced for it to become a black hole. The Sun’s mass is approximately
Solution
Although the Sun is not massive enough to become a black hole (its radius is approximately 700,000 kilometers), the Schwarzchild model theoretically indicates that if the Sun’s radius were reduced to approximately 2978 meters, that is, about its present size, it would become a black hole.
The speed of blood, S, in centimeters per second, located r centimeters from the central axis of an artery is modeled by
Find the speed of blood at the central axis of this artery.
Objective 3: Use scientific notation
Scientific Notation
Objective 3Use scientific notation.
Watch Video
Earth is a 4.5-billion-year-old ball of rock orbiting the Sun. Because a billion is (see Table P.3), the age of our world, in years, can be expressed as
The number is written in a form called scientific notation.
Table P.3 Names of Large Numbers
hundred
thousand
million
billion
trillion
quadrillion
quintillion
sextillion
septillion
octillion
nonillion
googol
googolplex
Scientific Notation
A number is written in scientific notation when it is expressed in the form
where the absolute value of a is greater than or equal to 1 and less than 10 and n is an integer.
It is customary to use the multiplication symbol, ×, rather than a dot, when writing a number in scientific notation.
Converting from Scientific to Decimal Notation
Here are two examples of numbers in scientific notation:
Do you see that the number with the positive exponent is relatively large and the number with the negative exponent is relatively small?
We can use n, the exponent on the 10 in to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the rightn places. If n is negative, move the decimal point in a to the left places.
Example 2 Converting from Scientific to Decimal Notation
Write each number in decimal notation:
Solution
In each case, we use the exponent on the 10 to determine how far to move the decimal point and in which direction. In parts (a) and (b), the exponent is positive, so we move the decimal point to the right. In parts (c) and (d), the exponent is negative, so we move the decimal point to the left.
To convert from decimal notation to scientific notation, we reverse the procedure of Example 2.
Converting from Decimal to Scientific Notation
Write the number in the form
Determine a, the numerical factor. Move the decimal point in the given number to obtain a number whose absolute value is between 1 and 10, including 1.
Determine n, the exponent on The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the decimal point was moved to the left, negative if the decimal point was moved to the right, and 0 if the decimal point was not moved.
Example 3 Converting from Decimal Notation to Scientific Notation
Perform the indicated computations, writing the answers in scientific notation:
Applications: Putting Numbers in Perspective
Due to tax cuts and spending increases, the United States began accumulating large deficits in the 1980s. To finance the deficit, the government had borrowed $25.5 trillion as of May 2020. The graph in Figure P.10 shows the national debt increasing over time. The amount shown for 2020 is midyear; the big increase in the national debt in the early part of the year was due in part to the economic impact of the COVID-19 pandemic.
Example 6 shows how we can use scientific notation to comprehend the meaning of a number such as 25.5 trillion.
Example 6 The National Debt
As of May 2020, the national debt was $25.5 trillion, or dollars. At that time, the U.S. population was approximately 331,000,000 (331 million), or If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay?
Solution
The amount each citizen must pay is the total debt, dollars, divided by the number of citizens,
Every U.S. citizen would have to pay approximately $77,000 to the federal government to pay off the national debt. At the end of 2019, the median yearly income of full-time workers in the United States was $48,672 (Source: Bureau of Labor Statistics); a worker with this income would have to pay more than 1.5 years’ salary to pay off the national debt.
At the end of 2010, the national debt was $13.5 trillion. At that time, the U.S. population was approximately 309 million. Write 13.5 trillion and 309 million in scientific notation. Then determine how much each citizen would have had to pay if the national debt in 2010 had been evenly divided among every individual in the 2010 population. Round to the nearest thousand dollars. Compare this amount to the $77,000 per person in 2020 from Example 6. How much did each individual’s share of the national debt increase between 2010 and 2020?
An Application: Black Holes in Space
The concept of a black hole, a region in space where matter appears to vanish, intrigues scientists and nonscientists alike. Scientists theorize that when massive stars run out of nuclear fuel, they begin to collapse under the force of their own gravity. As the star collapses, its density increases. In turn, the force of gravity increases so tremendously that even light cannot escape from the star. Consequently, it appears black.
A mathematical model, called the Schwarzchild formula, describes the critical value to which the radius of a massive body must be reduced for it to become a black hole. This model forms the basis of our next example.
Example 7 An Application of Scientific Notation
Use the Schwarzchild formula
where
to determine to what length the radius of the Sun must be reduced for it to become a black hole. The Sun’s mass is approximately
Solution
Although the Sun is not massive enough to become a black hole (its radius is approximately 700,000 kilometers), the Schwarzchild model theoretically indicates that if the Sun’s radius were reduced to approximately 2978 meters, that is, about its present size, it would become a black hole.
C8. A positive number is written in scientific notation when it is expressed in the form , where a is ________________________ and n is a/an ______.
C9. True or false: is written in scientific notation. ______
C10. True or false: is written in scientific notation. ______
P.2: Exercise Set
P.2 Exercise Set
Practice Exercises
Evaluate each exponential expression in Exercises 1–22.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Simplify each exponential expression in Exercises 23–64.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–76, write each number in decimal notation without the use of exponents.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
In Exercises 77–86, write each number in scientific notation.
77. 32,000
78. 64,000
79. 638,000,000,000,000,000
80. 579,000,000,000,000,000
81.
82.
83. 0.0027
84. 0.0083
85.
86.
In Exercises 87–106, perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
Practice PLUS
In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers.
107.
108.
109.
110.
111.
112.
113.
114.
Application Exercises
The graph shows the cost, in billions of dollars, and the enrollment, in millions of people, for various federal social programs for a recent year. Use the numbers shown to solve Exercises 115–117.
Express the federal cost for Social Security in scientific notation.
Express the enrollment in Social Security in scientific notation.
Use your answers from parts (a) and (b) to determine the average yearly per person benefit for Social Security. Round to the nearest dollar and express the answer in scientific notation and in decimal notation.
Use your decimal answer from part (c) to determine the average monthly per person benefit, expressed in decimal notation, for Social Security.
116.
Express the federal cost for the food stamps program in scientific notation.
Express the enrollment in the food stamps program in scientific notation.
Use your answers from parts (a) and (b) to determine the average yearly per person benefit for the food stamps program. Round to the nearest dollar and express the answer in scientific notation and in decimal notation.
Use your decimal answer from part (c) to determine the average monthly per person benefit, expressed in decimal notation, for the food stamps program.
117. Medicaid provides health insurance for the poor. Medicare provides health insurance for people 65 and older, as well as younger people who are disabled. Which program provides the greater yearly per person benefit? By how much, rounded to the nearest dollar?
We have seen that in May 2020 the U.S. national debt was $25.5 trillion. In Exercises 118–120, you will use scientific notation to put a number like 25.5 trillion in perspective.
118.
Express 25.5 trillion in scientific notation.
Each year, Americans spend $254 billion on summer vacations. Express this number in scientific notation.
Use your answers from parts (a) and (b) to determine how many years Americans can have free summer vacations for $25.5 trillion.
119.
Express 25.5 trillion in scientific notation.
Assume that four years of tuition, fees, and room and board at a public U.S. college cost approximately $60,000. Express this number in scientific notation.
Use your answers from parts (a) and (b) to determine how many Americans could receive a free college education for $25.5 trillion.
120. In 2019, the United States government spent more than it had collected in taxes, resulting in a budget deficit of $984 billion.
Express 984 billion in scientific notation.
There are approximately 32,000,000 seconds in a year. Write this number in scienfic notation.
Use your answers from parts (a) and (b) to determine approximately how many years is 984 billion seconds. (Note: 984 billion seconds would take us back in time to a period when Neanderthals were using stones to make tools.)
Explaining the Concepts
121. Describe what it means to raise a number to a power. In your description, include a discussion of the difference between and
122. Explain the product rule for exponents. Use in your explanation.
123. Explain the power rule for exponents. Use in your explanation.
124. Explain the quotient rule for exponents. Use in your explanation.
125. Why is not simplified? What must be done to simplify the expression?
126. How do you know if a number is written in scientific notation?
127. Explain how to convert from scientific to decimal notation and give an example.
128. Explain how to convert from decimal to scientific notation and give an example.
Critical Thinking Exercises
Make Sense? In Exercises 129–132, determine whether each statement makes sense or does not make sense, and explain your reasoning.
129. There are many exponential expressions that are equal to such as and
130. If is raised to the third power, the result is a number between 0 and 1.
131. The population of Colorado is approximately
132. I just finished reading a book that contained approximately words.
In Exercises 133–140, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
133.
134.
135.
136.
137.
138.
139.
140.
141. The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained.
142. If and what is the relationship among A, C, and D?
143. Our hearts beat approximately 70 times per minute. Express in scientific notation how many times the heart beats over a lifetime of 80 years. Round the decimal factor in your scientific notation answer to two decimal places.
Group Exercise
144. Putting Numbers into Perspective. A large number can be put into perspective by comparing it with another number. For example, we put the $25.5 trillion national debt in perspective (Example 6) by comparing this number to the number of U.S. citizens.
For this project, each group member should consult an almanac, a newspaper, or the Internet to find a number greater than one million. Explain to other members of the group the context in which the large number is used. Express the number in scientific notation. Then put the number into perspective by comparing it with another number.
Preview Exercises
Exercises 145–147 will help you prepare for the material covered in the next section.
145.
Find
Find
Based on your answers to parts (a) and (b), what can you conclude?
146.
Use a calculator to approximate to two decimal places.
Use a calculator to approximate to two decimal places.
Based on your answers to parts (a) and (b), what can you conclude?
147.
Simplify:
Simplify:
P.2: Exercise Set
P.2 Exercise Set
Practice Exercises
Evaluate each exponential expression in Exercises 1–22.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Simplify each exponential expression in Exercises 23–64.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–76, write each number in decimal notation without the use of exponents.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
In Exercises 77–86, write each number in scientific notation.
77. 32,000
78. 64,000
79. 638,000,000,000,000,000
80. 579,000,000,000,000,000
81.
82.
83. 0.0027
84. 0.0083
85.
86.
In Exercises 87–106, perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
Practice PLUS
In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers.
107.
108.
109.
110.
111.
112.
113.
114.
Application Exercises
The graph shows the cost, in billions of dollars, and the enrollment, in millions of people, for various federal social programs for a recent year. Use the numbers shown to solve Exercises 115–117.
Express the federal cost for Social Security in scientific notation.
Express the enrollment in Social Security in scientific notation.
Use your answers from parts (a) and (b) to determine the average yearly per person benefit for Social Security. Round to the nearest dollar and express the answer in scientific notation and in decimal notation.
Use your decimal answer from part (c) to determine the average monthly per person benefit, expressed in decimal notation, for Social Security.
116.
Express the federal cost for the food stamps program in scientific notation.
Express the enrollment in the food stamps program in scientific notation.
Use your answers from parts (a) and (b) to determine the average yearly per person benefit for the food stamps program. Round to the nearest dollar and express the answer in scientific notation and in decimal notation.
Use your decimal answer from part (c) to determine the average monthly per person benefit, expressed in decimal notation, for the food stamps program.
117. Medicaid provides health insurance for the poor. Medicare provides health insurance for people 65 and older, as well as younger people who are disabled. Which program provides the greater yearly per person benefit? By how much, rounded to the nearest dollar?
We have seen that in May 2020 the U.S. national debt was $25.5 trillion. In Exercises 118–120, you will use scientific notation to put a number like 25.5 trillion in perspective.
118.
Express 25.5 trillion in scientific notation.
Each year, Americans spend $254 billion on summer vacations. Express this number in scientific notation.
Use your answers from parts (a) and (b) to determine how many years Americans can have free summer vacations for $25.5 trillion.
119.
Express 25.5 trillion in scientific notation.
Assume that four years of tuition, fees, and room and board at a public U.S. college cost approximately $60,000. Express this number in scientific notation.
Use your answers from parts (a) and (b) to determine how many Americans could receive a free college education for $25.5 trillion.
120. In 2019, the United States government spent more than it had collected in taxes, resulting in a budget deficit of $984 billion.
Express 984 billion in scientific notation.
There are approximately 32,000,000 seconds in a year. Write this number in scienfic notation.
Use your answers from parts (a) and (b) to determine approximately how many years is 984 billion seconds. (Note: 984 billion seconds would take us back in time to a period when Neanderthals were using stones to make tools.)
Explaining the Concepts
121. Describe what it means to raise a number to a power. In your description, include a discussion of the difference between and
122. Explain the product rule for exponents. Use in your explanation.
123. Explain the power rule for exponents. Use in your explanation.
124. Explain the quotient rule for exponents. Use in your explanation.
125. Why is not simplified? What must be done to simplify the expression?
126. How do you know if a number is written in scientific notation?
127. Explain how to convert from scientific to decimal notation and give an example.
128. Explain how to convert from decimal to scientific notation and give an example.
Critical Thinking Exercises
Make Sense? In Exercises 129–132, determine whether each statement makes sense or does not make sense, and explain your reasoning.
129. There are many exponential expressions that are equal to such as and
130. If is raised to the third power, the result is a number between 0 and 1.
131. The population of Colorado is approximately
132. I just finished reading a book that contained approximately words.
In Exercises 133–140, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
133.
134.
135.
136.
137.
138.
139.
140.
141. The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained.
142. If and what is the relationship among A, C, and D?
143. Our hearts beat approximately 70 times per minute. Express in scientific notation how many times the heart beats over a lifetime of 80 years. Round the decimal factor in your scientific notation answer to two decimal places.
Group Exercise
144. Putting Numbers into Perspective. A large number can be put into perspective by comparing it with another number. For example, we put the $25.5 trillion national debt in perspective (Example 6) by comparing this number to the number of U.S. citizens.
For this project, each group member should consult an almanac, a newspaper, or the Internet to find a number greater than one million. Explain to other members of the group the context in which the large number is used. Express the number in scientific notation. Then put the number into perspective by comparing it with another number.
Preview Exercises
Exercises 145–147 will help you prepare for the material covered in the next section.
145.
Find
Find
Based on your answers to parts (a) and (b), what can you conclude?
146.
Use a calculator to approximate to two decimal places.
Use a calculator to approximate to two decimal places.
Based on your answers to parts (a) and (b), what can you conclude?
147.
Simplify:
Simplify:
Objective 1: Evaluate square roots
Square Roots
Objective 1Evaluate square roots.
Watch Video
From our earlier work with exponents, we are aware that the square of both 5 and is 25:
The reverse operation of squaring a number is finding the square root of a number. For example,
One square root of 25 is 5 because
Another square root of 25 is because
In general, ifthen b is a square root of a.
The symbol is used to denote the nonnegative or principal square root of a number. For example,
because and 5 is positive.
because and 10 is positive.
The symbol that we use to denote the principal square root is called a radical sign. The number under the radical sign is called the radicand. Together we refer to the radical sign and its radicand as a radical expression.
Definition of the Principal Square Root
If a is a nonnegative real number, the nonnegative number b such that denoted by is the principal square root of a.
The symbol is used to denote the negative square root of a number. For example,
A number that is the square of a rational number is called a perfect square. All the radicands in Example 1 and Check Point 1 are perfect squares.
64 is a perfect square because Thus,
is a perfect square because Thus,
Let’s see what happens to the radical expression if x is a negative number. Is the square root of a negative number a real number? For example, consider Is there a real number whose square is No. Thus, is not a real number. In general, a square root of a negative number is not a real number.
If a number a is nonnegative then For example,
Objective 1: Evaluate square roots
Square Roots
Objective 1Evaluate square roots.
Watch Video
From our earlier work with exponents, we are aware that the square of both 5 and is 25:
The reverse operation of squaring a number is finding the square root of a number. For example,
One square root of 25 is 5 because
Another square root of 25 is because
In general, ifthen b is a square root of a.
The symbol is used to denote the nonnegative or principal square root of a number. For example,
because and 5 is positive.
because and 10 is positive.
The symbol that we use to denote the principal square root is called a radical sign. The number under the radical sign is called the radicand. Together we refer to the radical sign and its radicand as a radical expression.
Definition of the Principal Square Root
If a is a nonnegative real number, the nonnegative number b such that denoted by is the principal square root of a.
The symbol is used to denote the negative square root of a number. For example,
A number that is the square of a rational number is called a perfect square. All the radicands in Example 1 and Check Point 1 are perfect squares.
64 is a perfect square because Thus,
is a perfect square because Thus,
Let’s see what happens to the radical expression if x is a negative number. Is the square root of a negative number a real number? For example, consider Is there a real number whose square is No. Thus, is not a real number. In general, a square root of a negative number is not a real number.
If a number a is nonnegative then For example,
Objective 3: Use the product rule to simplify square roots
The Product Rule for Square Roots
Objective 3Use the product rule to simplify square roots.
Watch Video
A rule for multiplying square roots can be generalized by comparing and Notice that
Because we obtain 10 in both situations, the original radical expressions must be equal. That is,
This result is a special case of the product rule for square roots that can be generalized as follows:
The Product Rule for Square Roots
If a and b represent nonnegative real numbers, then
A square root is simplified when its radicand has no factors other than 1 that are perfect squares. For example, is not simplified because it can be expressed as and 100 is a perfect square. Example 2 shows how the product rule is used to remove from the square root any perfect squares that occur as factors.
Example 2 Using the Product Rule to Simplify Square Roots
Simplify:
Solution
We can simplify using the product rule only if and represent nonnegative real numbers. Thus,
Two or more square roots can be combined using the distributive property provided that they have the same radicand. Such radicals are called like radicals. For example,
In some cases, radicals can be combined once they have been simplified. For example, to add and we can write as because 4 is a perfect square factor of 8.
Example 5 Combining Radicals That First Require Simplification
Figure P.11 The calculator screen shows approximate values for and
This process involves rewriting a radical expression as an equivalent expression in which the denominator no longer contains any radicals. The process is called rationalizing the denominator. If the denominator consists of the square root of a natural number that is not a perfect square, multiply the numerator and the denominator by the smallest number that produces the square root of a perfect square in the denominator.
Example 6 Rationalizing Denominators
Rationalize the denominator:
Solution
If we multiply the numerator and the denominator of by the denominator becomes Therefore, we multiply by 1, choosing for 1.
How can we rationalize a denominator if the denominator contains two terms with one or more square roots? Multiply the numerator and the denominator by the conjugate of the denominator. Here are three examples of such expressions:
The product of the denominator and its conjugate is found using the formula
The simplified product will not contain a radical.
Example 7 Rationalizing a Denominator Containing Two Terms
Rationalize the denominator:
Solution
The conjugate of the denominator is If we multiply the numerator and denominator by the simplified denominator will not contain a radical. Therefore, we multiply by 1, choosing for 1.
Figure P.11 The calculator screen shows approximate values for and
This process involves rewriting a radical expression as an equivalent expression in which the denominator no longer contains any radicals. The process is called rationalizing the denominator. If the denominator consists of the square root of a natural number that is not a perfect square, multiply the numerator and the denominator by the smallest number that produces the square root of a perfect square in the denominator.
Example 6 Rationalizing Denominators
Rationalize the denominator:
Solution
If we multiply the numerator and the denominator of by the denominator becomes Therefore, we multiply by 1, choosing for 1.
How can we rationalize a denominator if the denominator contains two terms with one or more square roots? Multiply the numerator and the denominator by the conjugate of the denominator. Here are three examples of such expressions:
The product of the denominator and its conjugate is found using the formula
The simplified product will not contain a radical.
Example 7 Rationalizing a Denominator Containing Two Terms
Rationalize the denominator:
Solution
The conjugate of the denominator is If we multiply the numerator and denominator by the simplified denominator will not contain a radical. Therefore, we multiply by 1, choosing for 1.
Objective 7: Evaluate and perform operations with higher roots
Other Kinds of Roots
Objective 7Evaluate and perform operations with higher roots.
Watch Video
We define the principal nth root of a real number a, symbolized by as follows:
Definition of the Principal nth Root of a Real Number
If n, the index, is even, then a is nonnegative and b is also nonnegative If n is odd, a and b can be any real numbers.
For example,
The same vocabulary that we learned for square roots applies to nth roots. The symbol is called a radical and the expression under the radical is called the radicand.
A number that is the nth power of a rational number is called a perfect nth power. For example, 8 is a perfect third power, or perfect cube, because Thus, In general, one of the following rules can be used to find the nth root of a perfect nth power:
Objective 7: Evaluate and perform operations with higher roots
Other Kinds of Roots
Objective 7Evaluate and perform operations with higher roots.
Watch Video
We define the principal nth root of a real number a, symbolized by as follows:
Definition of the Principal nth Root of a Real Number
If n, the index, is even, then a is nonnegative and b is also nonnegative If n is odd, a and b can be any real numbers.
For example,
The same vocabulary that we learned for square roots applies to nth roots. The symbol is called a radical and the expression under the radical is called the radicand.
A number that is the nth power of a rational number is called a perfect nth power. For example, 8 is a perfect third power, or perfect cube, because Thus, In general, one of the following rules can be used to find the nth root of a perfect nth power:
Objective 8: Understand and use rational exponents
Rational Exponents
Objective 8Understand and use rational exponents.
Watch Video
We define rational exponents so that their properties are the same as the properties for integer exponents. For example, we know that exponents are multiplied when an exponential expression is raised to a power. For this to be true,
We also know that
Can you see that the square of both and is 7? It is reasonable to conclude that
We can generalize the fact that means with the following definition:
The Definition of
If represents a real number, where is an integer, then
In Example 10 and Check Point 10, each rational exponent had a numerator of 1. If the numerator is some other integer, we still want to multiply exponents when raising a power to a power. For this reason,
Do you see that the denominator, 3, of the rational exponent is the same as the index of the radical? The numerator, 2, of the rational exponent serves as an exponent in each of the two radical forms. We generalize these ideas with the following definition:
The Definition of
If represents a real number and is a positive rational number, then
Also,
Furthermore, if is a nonzero real number, then
The first form of the definition of shown again below, involves taking the root first. This form is often preferable because smaller numbers are involved. Notice that the rational exponent consists of two parts, indicated by the following voice balloons:
Objective 8: Understand and use rational exponents
Rational Exponents
Objective 8Understand and use rational exponents.
Watch Video
We define rational exponents so that their properties are the same as the properties for integer exponents. For example, we know that exponents are multiplied when an exponential expression is raised to a power. For this to be true,
We also know that
Can you see that the square of both and is 7? It is reasonable to conclude that
We can generalize the fact that means with the following definition:
The Definition of
If represents a real number, where is an integer, then
In Example 10 and Check Point 10, each rational exponent had a numerator of 1. If the numerator is some other integer, we still want to multiply exponents when raising a power to a power. For this reason,
Do you see that the denominator, 3, of the rational exponent is the same as the index of the radical? The numerator, 2, of the rational exponent serves as an exponent in each of the two radical forms. We generalize these ideas with the following definition:
The Definition of
If represents a real number and is a positive rational number, then
Also,
Furthermore, if is a nonzero real number, then
The first form of the definition of shown again below, involves taking the root first. This form is often preferable because smaller numbers are involved. Notice that the rational exponent consists of two parts, indicated by the following voice balloons:
Objective 8: Understand and use rational exponents
Rational Exponents
Objective 8Understand and use rational exponents.
Watch Video
We define rational exponents so that their properties are the same as the properties for integer exponents. For example, we know that exponents are multiplied when an exponential expression is raised to a power. For this to be true,
We also know that
Can you see that the square of both and is 7? It is reasonable to conclude that
We can generalize the fact that means with the following definition:
The Definition of
If represents a real number, where is an integer, then
In Example 10 and Check Point 10, each rational exponent had a numerator of 1. If the numerator is some other integer, we still want to multiply exponents when raising a power to a power. For this reason,
Do you see that the denominator, 3, of the rational exponent is the same as the index of the radical? The numerator, 2, of the rational exponent serves as an exponent in each of the two radical forms. We generalize these ideas with the following definition:
The Definition of
If represents a real number and is a positive rational number, then
Also,
Furthermore, if is a nonzero real number, then
The first form of the definition of shown again below, involves taking the root first. This form is often preferable because smaller numbers are involved. Notice that the rational exponent consists of two parts, indicated by the following voice balloons:
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–44, add or subtract terms whenever possible.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
In Exercises 45–54, rationalize the denominator.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
Simplify the radical expressions in Exercises 67–74, if possible.
67.
68.
69.
70.
71.
72.
73.
74.
In Exercises 75–82, add or subtract terms whenever possible.
75.
76.
77.
78.
79.
80.
81.
82.
In Exercises 83–90, evaluate each expression without using a calculator.
83.
84.
85.
86.
87.
88.
89.
90.
In Exercises 91–100, simplify using properties of exponents.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
In Exercises 101–108, simplify by reducing the index of the radical.
101.
102.
103.
104.
105.
106.
107.
108.
Practice PLUS
In Exercises 109–110, evaluate each expression.
109.
110.
In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers.
111.
112.
113.
114.
Application Exercises
115. Do you expect to pay more taxes than were withheld? Would you be surprised to know that the percentage of taxpayers who receive a refund and the percentage of taxpayers who pay more taxes vary according to age? The formula
models the percentage, P, of taxpayers who are x years old who must pay more taxes.
What percentage of 25-year-olds must pay more taxes?
Rewrite the formula by rationalizing the denominator.
Use the rationalized form of the formula from part (b) to find the percentage of 25-year-olds who must pay more taxes. Do you get the same answer as you did in part (a)? If so, does this prove that you correctly rationalized the denominator? Explain.
116. America is getting older. The graph shows the elderly U.S. population for ages 65–84 and for ages 85 and older in 2010, with projections for 2020 and beyond.
The formula models the projected number of Americans ages 65–84, E, in millions, x years after 2010.
Use the formula to find the projected increase in the number of Americans ages 65–84, in millions, from 2020 to 2050. Express this difference in simplified radical form.
Use a calculator and write your answer in part (a) to the nearest tenth. Does this rounded decimal overestimate or underestimate the difference in the projected data shown by the bar graph? By how much?
117. The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is
The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed.
Rationalize the denominator of the given ratio, called the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.
118. Use Einstein’s special-relativity equation
described in the Blitzer Bonus here, to solve this exercise. You are moving at 90% of the speed of light. Substitute 0.9c for v, your velocity, in the equation. What is your aging rate, correct to two decimal places, relative to a friend on Earth? If you are gone for 44 weeks, approximately how many weeks have passed for your friend?
The perimeter, P, of a rectangle with length l and width w is given by the formula The area, A, is given by the formula In Exercises 119–120, use these formulas to find the perimeter and area of each rectangle. Express answers in simplified radical form. Remember that perimeter is measured in linear units, such as feet or meters, and area is measured in square units, such as square feet, or square meters,
119.
120.
Explaining the Concepts
121. Explain how to simplify
122. Explain how to add
123. Describe what it means to rationalize a denominator. Use both and in your explanation.
124. What difference is there in simplifying and
125. What does mean?
126. Describe the kinds of numbers that have rational fifth roots.
127. Why must a and b represent nonnegative numbers when we write Is it necessary to use this restriction in the case of Explain.
128. Read the Blitzer Bonus here. The future is now: You have the opportunity to explore the cosmos in a starship traveling near the speed of light. The experience will enable you to understand the mysteries of the universe in deeply personal ways, transporting you to unimagined levels of knowing and being. The downside: You return from your two-year journey to a futuristic world in which friends and loved ones are long gone. Do you explore space or stay here on Earth? What are the reasons for your choice?
Critical Thinking Exercises
Make Sense? In Exercises 129–132, determine whether each statement makes sense or does not make sense, and explain your reasoning.
129. The unlike radicals and remind me of the unlike terms 3x and 5y that cannot be combined by addition or subtraction.
130. Using my calculator, I determined that so 6 must be a seventh root of 279,936.
131. I simplified the terms of and then I was able to add the like radicals.
132. When I use the definition for I usually prefer to first raise a to the m power because smaller numbers are involved.
In Exercises 133–136, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
133.
134.
135. The cube root of is not a real number.
136.
In Exercises 137–138, fill in each box to make the statement true.
137.
138.
139. Find the exact value of without the use of a calculator.
140. Place the correct symbol, > or <, in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator.
141.
A mathematics professor recently purchased a birthday cake for her son with the inscription
How old is the son?
The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, “Hold on! It is your birthday, so why not take of the cake? I’ll eat half of what’s left over.” How much of the cake did the professor eat?
Preview Exercises
Exercises 142–144 will help you prepare for the material covered in the next section.
142. Multiply:
143. Use the distributive property to multiply:
144. Simplify and express the answer in descending powers of x:
P.3: Exercise Set
P.3 Exercise Set
Practice Exercises
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–44, add or subtract terms whenever possible.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
In Exercises 45–54, rationalize the denominator.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
Simplify the radical expressions in Exercises 67–74, if possible.
67.
68.
69.
70.
71.
72.
73.
74.
In Exercises 75–82, add or subtract terms whenever possible.
75.
76.
77.
78.
79.
80.
81.
82.
In Exercises 83–90, evaluate each expression without using a calculator.
83.
84.
85.
86.
87.
88.
89.
90.
In Exercises 91–100, simplify using properties of exponents.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
In Exercises 101–108, simplify by reducing the index of the radical.
101.
102.
103.
104.
105.
106.
107.
108.
Practice PLUS
In Exercises 109–110, evaluate each expression.
109.
110.
In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers.
111.
112.
113.
114.
Application Exercises
115. Do you expect to pay more taxes than were withheld? Would you be surprised to know that the percentage of taxpayers who receive a refund and the percentage of taxpayers who pay more taxes vary according to age? The formula
models the percentage, P, of taxpayers who are x years old who must pay more taxes.
What percentage of 25-year-olds must pay more taxes?
Rewrite the formula by rationalizing the denominator.
Use the rationalized form of the formula from part (b) to find the percentage of 25-year-olds who must pay more taxes. Do you get the same answer as you did in part (a)? If so, does this prove that you correctly rationalized the denominator? Explain.
116. America is getting older. The graph shows the elderly U.S. population for ages 65–84 and for ages 85 and older in 2010, with projections for 2020 and beyond.
The formula models the projected number of Americans ages 65–84, E, in millions, x years after 2010.
Use the formula to find the projected increase in the number of Americans ages 65–84, in millions, from 2020 to 2050. Express this difference in simplified radical form.
Use a calculator and write your answer in part (a) to the nearest tenth. Does this rounded decimal overestimate or underestimate the difference in the projected data shown by the bar graph? By how much?
117. The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is
The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed.
Rationalize the denominator of the given ratio, called the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.
118. Use Einstein’s special-relativity equation
described in the Blitzer Bonus here, to solve this exercise. You are moving at 90% of the speed of light. Substitute 0.9c for v, your velocity, in the equation. What is your aging rate, correct to two decimal places, relative to a friend on Earth? If you are gone for 44 weeks, approximately how many weeks have passed for your friend?
The perimeter, P, of a rectangle with length l and width w is given by the formula The area, A, is given by the formula In Exercises 119–120, use these formulas to find the perimeter and area of each rectangle. Express answers in simplified radical form. Remember that perimeter is measured in linear units, such as feet or meters, and area is measured in square units, such as square feet, or square meters,
119.
120.
Explaining the Concepts
121. Explain how to simplify
122. Explain how to add
123. Describe what it means to rationalize a denominator. Use both and in your explanation.
124. What difference is there in simplifying and
125. What does mean?
126. Describe the kinds of numbers that have rational fifth roots.
127. Why must a and b represent nonnegative numbers when we write Is it necessary to use this restriction in the case of Explain.
128. Read the Blitzer Bonus here. The future is now: You have the opportunity to explore the cosmos in a starship traveling near the speed of light. The experience will enable you to understand the mysteries of the universe in deeply personal ways, transporting you to unimagined levels of knowing and being. The downside: You return from your two-year journey to a futuristic world in which friends and loved ones are long gone. Do you explore space or stay here on Earth? What are the reasons for your choice?
Critical Thinking Exercises
Make Sense? In Exercises 129–132, determine whether each statement makes sense or does not make sense, and explain your reasoning.
129. The unlike radicals and remind me of the unlike terms 3x and 5y that cannot be combined by addition or subtraction.
130. Using my calculator, I determined that so 6 must be a seventh root of 279,936.
131. I simplified the terms of and then I was able to add the like radicals.
132. When I use the definition for I usually prefer to first raise a to the m power because smaller numbers are involved.
In Exercises 133–136, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
133.
134.
135. The cube root of is not a real number.
136.
In Exercises 137–138, fill in each box to make the statement true.
137.
138.
139. Find the exact value of without the use of a calculator.
140. Place the correct symbol, > or <, in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator.
141.
A mathematics professor recently purchased a birthday cake for her son with the inscription
How old is the son?
The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, “Hold on! It is your birthday, so why not take of the cake? I’ll eat half of what’s left over.” How much of the cake did the professor eat?
Preview Exercises
Exercises 142–144 will help you prepare for the material covered in the next section.
142. Multiply:
143. Use the distributive property to multiply:
144. Simplify and express the answer in descending powers of x:
Can that be Axl, your author’s yellow lab, sharing a special moment with a baby chick? And if it is (it is), what possible relevance can this have to polynomials? An answer is promised before you reach the Exercise Set. For now, we open the section by defining and describing polynomials.
How We Define Polynomials
More than 2 million people have tested their racial prejudice using an online version of the Implicit Association Test. Most groups’ average scores fall between “slight” and “moderate” bias, but the differences among age groups are intriguing.
In this section’s Exercise Set (Exercises 91 and 92), you will be working with a model that measures bias:
In this model, S represents the score on the Implicit Association Test. (Higher scores indicate stronger bias.) The variable x represents age group.
The algebraic expression that appears on the right side of the model is an example of a polynomial. A polynomial is a single term or the sum of two or more terms containing variables with whole-number exponents. This particular polynomial contains four terms. Equations containing polynomials are used in such diverse areas as science, business, medicine, psychology, and sociology. In this section, we review basic ideas about polynomials and their operations.
Objective 1: Understand the vocabulary of polynomials
How We Describe Polynomials
Objective 1Understand the vocabulary of polynomials.
Watch Video
Consider the polynomial
We can express as
The polynomial contains four terms. It is customary to write the terms in the order of descending powers of the variable. This is the standard form of a polynomial.
Some polynomials contain only one variable. Each term of such a polynomial in x is of the form . If , the degree of is n. For example, the degree of the term is 3.
The Degree of
If the degree of is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.
Here is an example of a polynomial and the degree of each of its four terms:
Notice that the exponent on x for the term 2x, meaning is understood to be 1. For this reason, the degree of 2x is 1. You can think of as thus, its degree is 0.
A polynomial is simplified when it contains no grouping symbols and no like terms. A simplified polynomial that has exactly one term is called a monomial. A binomial is a simplified polynomial that has two terms. A trinomial is a simplified polynomial with three terms. Simplified polynomials with four or more terms have no special names.
The degree of a polynomial is the greatest degree of all the terms of the polynomial. For example, is a binomial of degree 2 because the degree of the first term is 2, and the degree of the other term is less than 2. Also, is a trinomial of degree 5 because the degree of the first term is 5, and the degrees of the other terms are less than 5.
Up to now, we have used x to represent the variable in a polynomial. However, any letter can be used. For example,
is a polynomial (in x) of degree 5. Because there are three terms, the polynomial is a trinomial.
is a polynomial (in y) of degree 3. Because there are four terms, the polynomial has no special name.
is a polynomial (in z) of degree 7. Because there are two terms, the polynomial is a binomial.
We can tie together the threads of our discussion with the formal definition of a polynomial in one variable. In this definition, the coefficients of the terms are represented by (read “a sub n”), (read “a sub n minus 1”), and so on. The small letters to the lower right of each a are called subscripts and are not exponents. Subscripts are used to distinguish one constant from another when a large and undetermined number of such constants are needed.
Definition of a Polynomial in x
A polynomial inx is an algebraic expression of the form
where and are real numbers, and n is a nonnegative integer. The polynomial is of degree n, is the leading coefficient, and is the constant term.
Objective 3: Multiply polynomials
Multiplying Polynomials
Objective 3Multiply polynomials.
Watch Video
The product of two monomials is obtained by using properties of exponents. For example,
Furthermore, we can use the distributive property to multiply a monomial and a polynomial that is not a monomial. For example,
Objective 4: Use FOIL in polynomial multiplication
The Product of Two Binomials: FOIL
Objective 4Use FOIL in polynomial multiplication.
Watch Video
Frequently, we need to find the product of two binomials. One way to perform this multiplication is to distribute each term in the first binomial through the second binomial. For example, we can find the product of the binomials and as follows:
We can also find the product of and using a method called FOIL, which is based on our preceding work. Any two binomials can be quickly multiplied by using the FOIL method, in which F represents the product of the first terms in each binomial, O represents the product of the outside terms, I represents the product of the inside terms, and L represents the product of the last, or second, terms in each binomial. For example, we can use the FOIL method to find the product of the binomials and as follows:
Objective 6: Perform operations with polynomials in several variables
Polynomials in Several Variables
Objective 6Perform operations with polynomials in several variables.
Watch Video
A polynomial in two variables, x and y, contains the sum of one or more monomials in the form The constant, a, is the coefficient. The exponents, n and m, represent whole numbers. The degree of the monomial is
Here is an example of a polynomial in two variables:
The degree of a polynomial in two variables is the highest degree of all its terms. For the preceding polynomial, the degree is 6.
Polynomials containing two or more variables can be added, subtracted, and multiplied just like polynomials that contain only one variable. For example, we can add the monomials and as follows:
Fill in each blank so that the resulting statement is true.
C1. A polynomial is a single term or the sum of two or more terms containing variables with exponents that are ____ numbers.
C2. It is customary to write the terms of a polynomial in the order of descending powers of the variable. This is called the ______ form of a polynomial.
C3. A simplified polynomial that has exactly one term is called a/an ______.
C4. A simplified polynomial that has two terms is called a/an ______.
C5. A simplified polynomial that has three terms is called a/an ______.
C7. Polynomials are added by combining _____ terms.
C8. To multiply , use the _______ property to multiply each term of the trinomial _________ by the monomial _____.
C9. To multiply , begin by multiplying each term of by _____. Then multiply each term of by ___. Then combine _____ terms.
C10. When using the FOIL method to find , the product of the first terms is _____, the product of the outside terms is _____, the product of the inside terms is _____, and the product of the last terms is _____.
C11. _______. The product of the sum and difference of the same two terms is the square of the first term ______ the square of the second term.
C12. _______. The square of a binomial sum is the first term ______ plus 2 times the ___________ plus the last term ______.
C13. ______. The square of a binomial difference is the first term squared ______ 2 times the _______________ ______ the last term squared.
In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form.
1.
2.
3.
4.
In Exercises 5–8, find the degree of the polynomial.
5.
6.
7.
8.
In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
9.
10.
11.
12.
13.
14.
In Exercises 15–82, find each product.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
Practice PLUS
In Exercises 83–90, perform the indicated operation or operations.
83.
84.
85.
86.
87.
88.
89.
90.
Application Exercises
The bar graph shows the differences among age groups on the Implicit Association Test that measures levels of racial prejudice. Higher scores indicate stronger bias. Exercises 91–92 are based on the information in the graph. In these exercises, the possible values of the variable x are the group numbers shown in the voice balloons below the bars in the graph.
Source: The Race Implicit Association Test on the Project Implicit Demonstration Website
The data can be described by the following polynomial model of degree 3:
In this polynomial model, S represents the score on the Implicit Association Test for age group x. Simplify the model.
Use the simplified form of the model from part (a) to find the score on the Implicit Association Test for the group in the 45–54 age range. How well does the model describe the score displayed by the bar graph?
92.
The data can be described by the following polynomial model of degree 3:
In this polynomial model, S represents the score on the Implicit Association Test for age group x. Simplify the model.
Use the simplified form of the model from part (a) to find the score on the Implicit Association Test for the group in the 55–64 age range. How well does the model describe the score displayed by the bar graph?
The volume, V, of a rectangular solid with length l, width w, and height h is given by the formula In Exercises 93–94, use this formula to write a polynomial in standard form that models, or represents, the volume of the open box.
93.
94.
In Exercises 95–96, write a polynomial in standard form that models, or represents, the area of the shaded region.
95.
96.
Explaining the Concepts
97. What is a polynomial in x?
98. Explain how to subtract polynomials.
99. Explain how to multiply two binomials using the FOIL method. Give an example with your explanation.
100. Explain how to find the product of the sum and difference of two terms. Give an example with your explanation.
101. Explain how to square a binomial difference. Give an example with your explanation.
102. Explain how to find the degree of a polynomial in two variables.
Critical Thinking Exercises
Make Sense? In Exercises 103–106, determine whether each statement makes sense or does not make sense, and explain your reasoning.
103. Knowing the difference between factors and terms is important: In I can distribute the exponent 2 on each factor, but in I cannot do the same thing on each term.
104. I used the FOIL method to find the product of and
105. Many English words have prefixes with meanings similar to those used to describe polynomials, such as monologue, binocular, and tricuspid.
106. Special-product formulas have patterns that make their multiplications quicker than using the FOIL method.
107. Express the area of the plane figure shown as a polynomial in standard form.
In Exercises 108–109, represent the volume of each figure as a polynomial in standard form.
108.
109.
110. Simplify:
Preview Exercises
Exercises 111–113 will help you prepare for the material covered in the next section. In each exercise, replace the boxed question mark with an integer that results in the given product. Some trial and error may be necessary.
111.
112.
113.
P.4: Exercise Set
P.4 Exercise Set
Practice Exercises
In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form.
1.
2.
3.
4.
In Exercises 5–8, find the degree of the polynomial.
5.
6.
7.
8.
In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
9.
10.
11.
12.
13.
14.
In Exercises 15–82, find each product.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
Practice PLUS
In Exercises 83–90, perform the indicated operation or operations.
83.
84.
85.
86.
87.
88.
89.
90.
Application Exercises
The bar graph shows the differences among age groups on the Implicit Association Test that measures levels of racial prejudice. Higher scores indicate stronger bias. Exercises 91–92 are based on the information in the graph. In these exercises, the possible values of the variable x are the group numbers shown in the voice balloons below the bars in the graph.
Source: The Race Implicit Association Test on the Project Implicit Demonstration Website
The data can be described by the following polynomial model of degree 3:
In this polynomial model, S represents the score on the Implicit Association Test for age group x. Simplify the model.
Use the simplified form of the model from part (a) to find the score on the Implicit Association Test for the group in the 45–54 age range. How well does the model describe the score displayed by the bar graph?
92.
The data can be described by the following polynomial model of degree 3:
In this polynomial model, S represents the score on the Implicit Association Test for age group x. Simplify the model.
Use the simplified form of the model from part (a) to find the score on the Implicit Association Test for the group in the 55–64 age range. How well does the model describe the score displayed by the bar graph?
The volume, V, of a rectangular solid with length l, width w, and height h is given by the formula In Exercises 93–94, use this formula to write a polynomial in standard form that models, or represents, the volume of the open box.
93.
94.
In Exercises 95–96, write a polynomial in standard form that models, or represents, the area of the shaded region.
95.
96.
Explaining the Concepts
97. What is a polynomial in x?
98. Explain how to subtract polynomials.
99. Explain how to multiply two binomials using the FOIL method. Give an example with your explanation.
100. Explain how to find the product of the sum and difference of two terms. Give an example with your explanation.
101. Explain how to square a binomial difference. Give an example with your explanation.
102. Explain how to find the degree of a polynomial in two variables.
Critical Thinking Exercises
Make Sense? In Exercises 103–106, determine whether each statement makes sense or does not make sense, and explain your reasoning.
103. Knowing the difference between factors and terms is important: In I can distribute the exponent 2 on each factor, but in I cannot do the same thing on each term.
104. I used the FOIL method to find the product of and
105. Many English words have prefixes with meanings similar to those used to describe polynomials, such as monologue, binocular, and tricuspid.
106. Special-product formulas have patterns that make their multiplications quicker than using the FOIL method.
107. Express the area of the plane figure shown as a polynomial in standard form.
In Exercises 108–109, represent the volume of each figure as a polynomial in standard form.
108.
109.
110. Simplify:
Preview Exercises
Exercises 111–113 will help you prepare for the material covered in the next section. In each exercise, replace the boxed question mark with an integer that results in the given product. Some trial and error may be necessary.
A two-year-old boy is asked, “Do you have a brother?” He answers, “Yes.” “What is your brother’s name?” “Tom.” Asked if Tom has a brother, the two-year-old replies, “No.” The child can go in the direction from self to brother, but he cannot reverse this direction and move from brother back to self.
As our intellects develop, we learn to reverse the direction of our thinking. Reversibility of thought is found throughout algebra. For example, we can multiply polynomials and show that
We can also reverse this process and express the resulting polynomial as
Factoring a polynomial expressed as the sum of monomials means finding an equivalent expression that is a product.
In this section, we will be factoring over the set of integers, meaning that the coefficients in the factors are integers. Polynomials that cannot be factored using integer coefficients are called irreducible over the integers, or prime.
The goal in factoring a polynomial is to use one or more factoring techniques until each of the polynomial’s factors, except possibly for a monomial factor, is prime or irreducible. In this situation, the polynomial is said to be factored completely.
We will now discuss basic techniques for factoring polynomials.
Objective 2: Factor by grouping
Factoring by Grouping
Objective 2Factor by grouping.
Watch Video
Some polynomials have only a greatest common factor of 1. However, by a suitable grouping of the terms, it still may be possible to factor. This process, called factoring by grouping, is illustrated in Example 2.
Example 2 Factoring by Grouping
Factor:
Solution
There is no factor other than 1 common to all terms. However, we can group terms that have a common factor:
We now factor the given polynomial as follows:
Thus, Check the factorization by multiplying the right side of the equation using the FOIL method. Because the factorization is correct, you should obtain the original polynomial.
If no such combination exists, the polynomial is prime.
Example 3 Factoring a Trinomial Whose Leading Coefficient Is 1
Factor:
Solution
Step 1 FIND TWO FIRST TERMS WHOSE PRODUCT IS
Step 1 (REPEATED)
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS 8.
Factors of 8
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of the possible factorizations:
In factoring a trinomial of the form you can speed things up by listing the factors of c and then finding their sums. We are interested in a sum of b. For example, in factoring we are interested in the factors of 8 whose sum is 6.
Example 5 Factoring a Trinomial Whose Leading Coefficient Is Not 1
Factor:
Solution
Step 1 FIND TWO FIRST TERMS WHOSE PRODUCT IS
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS The possible factorizations are and
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of the possible factorizations:
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS The possible factorizations are and
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of possible factorizations:
If no such combination exists, the polynomial is prime.
Example 3 Factoring a Trinomial Whose Leading Coefficient Is 1
Factor:
Solution
Step 1 FIND TWO FIRST TERMS WHOSE PRODUCT IS
Step 1 (REPEATED)
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS 8.
Factors of 8
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of the possible factorizations:
In factoring a trinomial of the form you can speed things up by listing the factors of c and then finding their sums. We are interested in a sum of b. For example, in factoring we are interested in the factors of 8 whose sum is 6.
Example 5 Factoring a Trinomial Whose Leading Coefficient Is Not 1
Factor:
Solution
Step 1 FIND TWO FIRST TERMS WHOSE PRODUCT IS
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS The possible factorizations are and
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of the possible factorizations:
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS The possible factorizations are and
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of possible factorizations:
If no such combination exists, the polynomial is prime.
Example 3 Factoring a Trinomial Whose Leading Coefficient Is 1
Factor:
Solution
Step 1 FIND TWO FIRST TERMS WHOSE PRODUCT IS
Step 1 (REPEATED)
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS 8.
Factors of 8
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of the possible factorizations:
In factoring a trinomial of the form you can speed things up by listing the factors of c and then finding their sums. We are interested in a sum of b. For example, in factoring we are interested in the factors of 8 whose sum is 6.
Example 5 Factoring a Trinomial Whose Leading Coefficient Is Not 1
Factor:
Solution
Step 1 FIND TWO FIRST TERMS WHOSE PRODUCT IS
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS The possible factorizations are and
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of the possible factorizations:
Step 2 FIND TWO LAST TERMS WHOSE PRODUCT IS The possible factorizations are and
Step 3 TRY VARIOUS COMBINATIONS OF THESE FACTORS. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to Here is a list of possible factorizations:
We have seen that a polynomial is factored completely when it is written as the product of prime polynomials. To be sure that you have factored completely, check to see whether any factors with more than one term in the factored polynomial can be factored further. If so, continue factoring.
Our next factoring technique is obtained by reversing the special products for squaring binomials. The trinomials that are factored using this technique are called perfect square trinomials.
Factoring Perfect Square Trinomials
Let A and B be real numbers, variables, or algebraic expressions.
The two items in the box show that perfect square trinomials, and come in two forms: one in which the coefficient of the middle term is positive and one in which the coefficient of the middle term is negative. Here’s how to recognize a perfect square trinomial:
The first and last terms are squares of monomials or integers.
The middle term is twice the product of the expressions being squared in the first and last terms.
Objective 7: Use a general strategy for factoring polynomials
A Strategy for Factoring Polynomials
Objective 7Use a general strategy for factoring polynomials.
Watch Video
It is important to practice factoring a wide variety of polynomials so that you can quickly select the appropriate technique. The polynomial is factored completely when all its polynomial factors, except possibly for monomial factors, are prime. Because of the commutative property, the order of the factors does not matter.
A Strategy for Factoring a Polynomial
If there is a common factor, factor out the GCF.
Determine the number of terms in the polynomial and try factoring as follows:
If there are two terms, can the binomial be factored by using one of the following special forms?
If there are three terms, is the trinomial a perfect square trinomial? If so, factor by using one of the following special forms:
If the trinomial is not a perfect square trinomial, try factoring by trial and error.
If there are four or more terms, try factoring by grouping.
Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely.
Example 11 Factoring a Polynomial
Factor:
Solution
Step 1 IF THERE IS A COMMON FACTOR, FACTOR OUT THE GCF. Because 2x is common to all terms, we factor it out.
Step 2 DETERMINE THE NUMBER OF TERMS AND FACTOR ACCORDINGLY. The factor has three terms and is a perfect square trinomial. We factor using
Step 1 IF THERE IS A COMMON FACTOR, FACTOR OUT THE GCF. Other than 1 or there is no common factor.
Step 2 DETERMINE THE NUMBER OF TERMS AND FACTOR ACCORDINGLY. There are four terms. We try factoring by grouping. It can be shown that grouping into two groups of two terms does not result in a common binomial factor. Let’s try grouping as a difference of squares.
Step 3 CHECK TO SEE IF FACTORS CAN BE FACTORED FURTHER. In this case, they cannot, so we have factored completely.
Objective 7: Use a general strategy for factoring polynomials
A Strategy for Factoring Polynomials
Objective 7Use a general strategy for factoring polynomials.
Watch Video
It is important to practice factoring a wide variety of polynomials so that you can quickly select the appropriate technique. The polynomial is factored completely when all its polynomial factors, except possibly for monomial factors, are prime. Because of the commutative property, the order of the factors does not matter.
A Strategy for Factoring a Polynomial
If there is a common factor, factor out the GCF.
Determine the number of terms in the polynomial and try factoring as follows:
If there are two terms, can the binomial be factored by using one of the following special forms?
If there are three terms, is the trinomial a perfect square trinomial? If so, factor by using one of the following special forms:
If the trinomial is not a perfect square trinomial, try factoring by trial and error.
If there are four or more terms, try factoring by grouping.
Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely.
Example 11 Factoring a Polynomial
Factor:
Solution
Step 1 IF THERE IS A COMMON FACTOR, FACTOR OUT THE GCF. Because 2x is common to all terms, we factor it out.
Step 2 DETERMINE THE NUMBER OF TERMS AND FACTOR ACCORDINGLY. The factor has three terms and is a perfect square trinomial. We factor using
Step 1 IF THERE IS A COMMON FACTOR, FACTOR OUT THE GCF. Other than 1 or there is no common factor.
Step 2 DETERMINE THE NUMBER OF TERMS AND FACTOR ACCORDINGLY. There are four terms. We try factoring by grouping. It can be shown that grouping into two groups of two terms does not result in a common binomial factor. Let’s try grouping as a difference of squares.
Step 3 CHECK TO SEE IF FACTORS CAN BE FACTORED FURTHER. In this case, they cannot, so we have factored completely.
In Exercises 1–10, factor out the greatest common factor.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–16, factor by grouping.
11.
12.
13.
14.
15.
16.
In Exercises 17–38, factor each trinomial, or state that the trinomial is prime.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
In Exercises 39–48, factor the difference of two squares.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–56, factor each perfect square trinomial.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–64, factor using the formula for the sum or difference of two cubes.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–92, factor completely, or state that the polynomial is prime.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
In Exercises 93–102, factor and simplify each algebraic expression.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
Practice PLUS
In Exercises 103–114, factor completely.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
Application Exercises
115. Your electronics store is having an incredible sale. The price on one laptop is reduced by 40%. Then the sale price is reduced by another 40%. If x is the laptop’s original price, the sale price can be modeled by
Factor out from each term. Then simplify the resulting expression.
Use the simplified expression from part (a) to answer these questions. With a 40% reduction followed by a 40% reduction, is the laptop selling at 20% of its original price? If not, at what percentage of the original price is it selling?
116. Your local electronics store is having an end-of-the-year sale. The price on an Ultra 4K HD television had been reduced by 30%. Now the sale price is reduced by another 30%. If x is the television’s original price, the sale price can be modeled by
Factor out from each term. Then simplify the resulting expression.
Use the simplified expression from part (a) to answer these questions. With a 30% reduction followed by a 30% reduction, is the television selling at 40% of its original price? If not, at what percentage of the original price is it selling?
In Exercises 117–120,
Write an expression for the area of the shaded region.
In Exercises 121–122, find the formula for the volume of the region outside the smaller rectangular solid and inside the larger rectangular solid. Then express the volume in factored form.
121.
122.
Explaining the Concepts
123. Using an example, explain how to factor out the greatest common factor of a polynomial.
124. Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.
125. Explain how to factor
126. Explain how to factor the difference of two squares. Provide an example with your explanation.
127. What is a perfect square trinomial and how is it factored?
128. Explain how to factor
129. What does it mean to factor completely?
Critical Thinking Exercises
Make Sense? In Exercises 130–133, determine whether each statement makes sense or does not make sense, and explain your reasoning.
130. Although appears in both and I’ll need to factor in different ways to obtain each polynomial’s factorization.
131. You grouped the polynomial’s terms using different groupings than I did, yet we both obtained the same factorization.
132. I factored completely and obtained
133. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.
In Exercises 134–137, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
134. is factored completely as
135. The trinomial is a prime polynomial.
136.
137.
In Exercises 138–141, factor completely.
138.
139.
140.
141.
In Exercises 142–143, find all integers b so that the trinomial can be factored.
142.
143.
Preview Exercises
Exercises 144–146 will help you prepare for the material covered in the next section.
144. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator.
In Exercises 145–146, perform the indicated operation. Where possible, reduce the answer to its lowest terms.
145.
146.
P.5: Exercise Set
P.5 Exercise Set
Practice Exercises
In Exercises 1–10, factor out the greatest common factor.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–16, factor by grouping.
11.
12.
13.
14.
15.
16.
In Exercises 17–38, factor each trinomial, or state that the trinomial is prime.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
In Exercises 39–48, factor the difference of two squares.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–56, factor each perfect square trinomial.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–64, factor using the formula for the sum or difference of two cubes.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–92, factor completely, or state that the polynomial is prime.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
In Exercises 93–102, factor and simplify each algebraic expression.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
Practice PLUS
In Exercises 103–114, factor completely.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
Application Exercises
115. Your electronics store is having an incredible sale. The price on one laptop is reduced by 40%. Then the sale price is reduced by another 40%. If x is the laptop’s original price, the sale price can be modeled by
Factor out from each term. Then simplify the resulting expression.
Use the simplified expression from part (a) to answer these questions. With a 40% reduction followed by a 40% reduction, is the laptop selling at 20% of its original price? If not, at what percentage of the original price is it selling?
116. Your local electronics store is having an end-of-the-year sale. The price on an Ultra 4K HD television had been reduced by 30%. Now the sale price is reduced by another 30%. If x is the television’s original price, the sale price can be modeled by
Factor out from each term. Then simplify the resulting expression.
Use the simplified expression from part (a) to answer these questions. With a 30% reduction followed by a 30% reduction, is the television selling at 40% of its original price? If not, at what percentage of the original price is it selling?
In Exercises 117–120,
Write an expression for the area of the shaded region.
In Exercises 121–122, find the formula for the volume of the region outside the smaller rectangular solid and inside the larger rectangular solid. Then express the volume in factored form.
121.
122.
Explaining the Concepts
123. Using an example, explain how to factor out the greatest common factor of a polynomial.
124. Suppose that a polynomial contains four terms. Explain how to use factoring by grouping to factor the polynomial.
125. Explain how to factor
126. Explain how to factor the difference of two squares. Provide an example with your explanation.
127. What is a perfect square trinomial and how is it factored?
128. Explain how to factor
129. What does it mean to factor completely?
Critical Thinking Exercises
Make Sense? In Exercises 130–133, determine whether each statement makes sense or does not make sense, and explain your reasoning.
130. Although appears in both and I’ll need to factor in different ways to obtain each polynomial’s factorization.
131. You grouped the polynomial’s terms using different groupings than I did, yet we both obtained the same factorization.
132. I factored completely and obtained
133. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.
In Exercises 134–137, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
134. is factored completely as
135. The trinomial is a prime polynomial.
136.
137.
In Exercises 138–141, factor completely.
138.
139.
140.
141.
In Exercises 142–143, find all integers b so that the trinomial can be factored.
142.
143.
Preview Exercises
Exercises 144–146 will help you prepare for the material covered in the next section.
144. Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator.
In Exercises 145–146, perform the indicated operation. Where possible, reduce the answer to its lowest terms.
145.
146.
Chapter P: Mid-Chapter Check Point
Chapter P Mid-Chapter Check Point
What You Know: We defined the real numbers and graphed them as points on a number line. We reviewed the basic rules of algebra, using these properties to simplify algebraic expressions. We expanded our knowledge of exponents to include exponents other than natural numbers:
We used properties of exponents to simplify exponential expressions and properties of radicals to simplify radical expressions. Finally, we performed operations with polynomials. We used a number of fast methods for finding products of polynomials, including the FOIL method for multiplying binomials, a special-product formula for the product of the sum and difference of two terms and special-product formulas for squaring binomials We reversed the direction of these formulas and reviewed how to factor polynomials. We used a general strategy, summarized in the box here, for factoring a wide variety of polynomials.
In Exercises 1–27, simplify the given expression or perform the indicated operation (and simplify, if possible), whichever is appropriate.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21. (Express the answer in scientific notation.)
22.
23.
24.
25.
26.
27.
In Exercises 28–34, factor completely, or state that the polynomial is prime.
28.
29.
30.
31.
32.
33.
34.
In Exercises 35–36, factor and simplify each algebraic expression.
35.
36.
37. List all the rational numbers in this set:
In Exercises 38–39, rewrite each expression without absolute value bars.
38.
39.
40. If the population of the United States is approximately and each person produces about 5 pounds of garbage per day, express the total number of pounds of garbage produced in the United States in one day in scientific notation.
41. A human brain contains neurons and a gorilla brain contains neurons. How many times as many neurons are in the brain of a human as in the brain of a gorilla?
42. College students are graduating with the highest debt burden in history, but that debt may finally be leveling off. The bar graph shows the average student loan debt in the United States for seven selected graduating classes from 2001 through 2019.
We can all do our part to help keep our beaches and riverbanks free of litter and other debris, but what happens when an industrial accident causes toxic chemicals to be discharged into our rivers or an oil spill covers our beaches? The costs of cleaning up after these environmental disasters can be enormous. In this section, we will model such costs using quotients of polynomials.
For example, the algebraic expression
describes the cost, in millions of dollars, to remove x percent of the pollutants that are discharged into a river. Removing a modest percentage of pollutants, say 40%, is far less costly than removing a substantially greater percentage, such as 95%. We see this by evaluating the algebraic expression for and
The cost increases from approximately $167 million to a possibly prohibitive $4750 million, or $4.75 billion. Costs spiral upward as the percentage of removed pollutants increases.
While cleanup costs vary based on the size of the disaster, consider this: As of April 2020, ten years after the explosion that caused the Deep Horizon oil spill in the Gulf of Mexico, BP (formerly The British Petroleum Company) had already spent nearly $70 billion on cleanup and settling claims.
Many algebraic expressions that describe costs of environmental projects are examples of rational expressions. First we will define rational expressions. Then we will review how to perform operations with such expressions.
Objective 2: Simplify rational expressions
Simplifying Rational Expressions
Objective 2Simplify rational expressions.
Watch Video
A rational expression is simplified if its numerator and denominator have no common factors other than 1 or The following procedure can be used to simplify rational expressions:
Simplifying Rational Expressions
Factor the numerator and the denominator completely.
Divide both the numerator and the denominator by any common factors.
Objective 5: Add and subtract rational expressions
Adding and Subtracting Rational Expressions with the Same Denominator
Objective 5Add and subtract rational expressions.
Watch Video
We add or subtract rational expressions with the same denominator by (1) adding or subtracting the numerators, (2) placing this result over the common denominator, and (3) simplifying, if possible.
Example 5 Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Different Denominators
We can gain insight into adding rational expressions with different denominators by looking closely at what we do when adding fractions with different denominators. For example, suppose that we want to add and We must first write the fractions with the same denominator. We look for the smallest number that contains both 2 and 3 as factors. This number, 6, is then used as the least common denominator, or LCD.
The least common denominator, or LCD, of several rational expressions is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
Finding the Least Common Denominator
Factor each denominator completely.
List the factors of the first denominator.
Add to the list in step 2 any factors of the second denominator that do not appear in the list.
Form the product of each different factor from the list in step 3. This product is the least common denominator.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of the second denominator is already in our list. That factor is 1. However, the other factor, , is not listed in step 2. We add to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus
In Example 6 and Check Point 6, the denominators do not have any common factors other than 1 and . Thus, the LCD is simply the product of the denominators. However, in Example 7, the denominators both have a factor of .
Example 7 Finding the Least Common Denominator
Find the LCD of
Solution
Step 1 FACTOR EACH DENOMINATOR COMPLETELY.
Step 2 LIST THE FACTORS OF THE FIRST DENOMINATOR.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of is already in our list. That factor is However, the other factor of is not listed in step 2. We add a second factor of to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus,
is the least common denominator.
In Example 7, if you were to multiply the denominators as given, you would get a common denominator, but not the LCD. While any common denominator can be used to add or subtract rational expressions, using the LCD greatly reduces the amount of work involved.
Finding the least common denominator for two (or more) rational expressions is the first step needed to add or subtract the expressions.
Adding and Subtracting Rational Expressions That Have Different Denominators
Find the LCD of the rational expressions.
Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
Add or subtract numerators, placing the resulting expression over the LCD.
If possible, simplify the resulting rational expression.
Example 8 Adding and Subtracting Rational Expressions with Different Denominators
Add or subtract:
.
Solution
Note that each of these denominators has only one term.
The second rational expression in the sum, , is undefined for , so .
Step 1 FIND THE LEAST COMMON DENOMINATOR. The denominators are 3 and x, which have no common factors. Their product, 3x, is the LCD.
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of 3x. We do so by multiplying the numerator and denominator of by x and multiplying the numerator and denominator of by 3.
Because and , we are not changing the value of either rational expression. Now we are ready to perform the addition.
Step 3 ADD NUMERATORS, PLACING THIS SUM OVER THE LCD.
Step 4 Simplify, if necessary. Because the numerator is prime, no further simplification is possible.
Both rational expressions in the subtraction are undefined for so .
Step 1 Find the least common denominator. The denominators are 6x and . Begin by listing the factors of 6x:
The factors of are 3, 3, x, and x. The list of factors of 6x only contains one factor of 3 and one factor of x, so we add one more 3 and one more x to the list:
The LCD is the product of the factors in this list. The LCD is .
Step 2 Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of . We do so by multiplying the numerator and denominator of by 3x and multiplying the numerator and denominator of by 2.
Example 9 Subtracting Rational Expressions with Different Denominators
Subtract: .
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. In Example 6, we found that the LCD for these rational expressions is .
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of . We do so by multiplying both the numerator and the denominator of each rational expression by any factor needed to convert the expression’s denominator into the LCD.
Example 10 Adding Rational Expressions with Different Denominators
Add:
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. Start by factoring the denominators.
The factors of the first denominator are and The only factor from the second denominator that is not listed is Thus, the least common denominator is
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of We do so by multiplying both the numerator and the denominator of each rational expression by any factor(s) needed to convert the expression’s denominator into the LCD.
Objective 5: Add and subtract rational expressions
Adding and Subtracting Rational Expressions with the Same Denominator
Objective 5Add and subtract rational expressions.
Watch Video
We add or subtract rational expressions with the same denominator by (1) adding or subtracting the numerators, (2) placing this result over the common denominator, and (3) simplifying, if possible.
Example 5 Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Different Denominators
We can gain insight into adding rational expressions with different denominators by looking closely at what we do when adding fractions with different denominators. For example, suppose that we want to add and We must first write the fractions with the same denominator. We look for the smallest number that contains both 2 and 3 as factors. This number, 6, is then used as the least common denominator, or LCD.
The least common denominator, or LCD, of several rational expressions is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
Finding the Least Common Denominator
Factor each denominator completely.
List the factors of the first denominator.
Add to the list in step 2 any factors of the second denominator that do not appear in the list.
Form the product of each different factor from the list in step 3. This product is the least common denominator.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of the second denominator is already in our list. That factor is 1. However, the other factor, , is not listed in step 2. We add to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus
In Example 6 and Check Point 6, the denominators do not have any common factors other than 1 and . Thus, the LCD is simply the product of the denominators. However, in Example 7, the denominators both have a factor of .
Example 7 Finding the Least Common Denominator
Find the LCD of
Solution
Step 1 FACTOR EACH DENOMINATOR COMPLETELY.
Step 2 LIST THE FACTORS OF THE FIRST DENOMINATOR.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of is already in our list. That factor is However, the other factor of is not listed in step 2. We add a second factor of to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus,
is the least common denominator.
In Example 7, if you were to multiply the denominators as given, you would get a common denominator, but not the LCD. While any common denominator can be used to add or subtract rational expressions, using the LCD greatly reduces the amount of work involved.
Finding the least common denominator for two (or more) rational expressions is the first step needed to add or subtract the expressions.
Adding and Subtracting Rational Expressions That Have Different Denominators
Find the LCD of the rational expressions.
Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
Add or subtract numerators, placing the resulting expression over the LCD.
If possible, simplify the resulting rational expression.
Example 8 Adding and Subtracting Rational Expressions with Different Denominators
Add or subtract:
.
Solution
Note that each of these denominators has only one term.
The second rational expression in the sum, , is undefined for , so .
Step 1 FIND THE LEAST COMMON DENOMINATOR. The denominators are 3 and x, which have no common factors. Their product, 3x, is the LCD.
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of 3x. We do so by multiplying the numerator and denominator of by x and multiplying the numerator and denominator of by 3.
Because and , we are not changing the value of either rational expression. Now we are ready to perform the addition.
Step 3 ADD NUMERATORS, PLACING THIS SUM OVER THE LCD.
Step 4 Simplify, if necessary. Because the numerator is prime, no further simplification is possible.
Both rational expressions in the subtraction are undefined for so .
Step 1 Find the least common denominator. The denominators are 6x and . Begin by listing the factors of 6x:
The factors of are 3, 3, x, and x. The list of factors of 6x only contains one factor of 3 and one factor of x, so we add one more 3 and one more x to the list:
The LCD is the product of the factors in this list. The LCD is .
Step 2 Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of . We do so by multiplying the numerator and denominator of by 3x and multiplying the numerator and denominator of by 2.
Example 9 Subtracting Rational Expressions with Different Denominators
Subtract: .
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. In Example 6, we found that the LCD for these rational expressions is .
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of . We do so by multiplying both the numerator and the denominator of each rational expression by any factor needed to convert the expression’s denominator into the LCD.
Example 10 Adding Rational Expressions with Different Denominators
Add:
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. Start by factoring the denominators.
The factors of the first denominator are and The only factor from the second denominator that is not listed is Thus, the least common denominator is
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of We do so by multiplying both the numerator and the denominator of each rational expression by any factor(s) needed to convert the expression’s denominator into the LCD.
Objective 5: Add and subtract rational expressions
Adding and Subtracting Rational Expressions with the Same Denominator
Objective 5Add and subtract rational expressions.
Watch Video
We add or subtract rational expressions with the same denominator by (1) adding or subtracting the numerators, (2) placing this result over the common denominator, and (3) simplifying, if possible.
Example 5 Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Different Denominators
We can gain insight into adding rational expressions with different denominators by looking closely at what we do when adding fractions with different denominators. For example, suppose that we want to add and We must first write the fractions with the same denominator. We look for the smallest number that contains both 2 and 3 as factors. This number, 6, is then used as the least common denominator, or LCD.
The least common denominator, or LCD, of several rational expressions is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
Finding the Least Common Denominator
Factor each denominator completely.
List the factors of the first denominator.
Add to the list in step 2 any factors of the second denominator that do not appear in the list.
Form the product of each different factor from the list in step 3. This product is the least common denominator.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of the second denominator is already in our list. That factor is 1. However, the other factor, , is not listed in step 2. We add to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus
In Example 6 and Check Point 6, the denominators do not have any common factors other than 1 and . Thus, the LCD is simply the product of the denominators. However, in Example 7, the denominators both have a factor of .
Example 7 Finding the Least Common Denominator
Find the LCD of
Solution
Step 1 FACTOR EACH DENOMINATOR COMPLETELY.
Step 2 LIST THE FACTORS OF THE FIRST DENOMINATOR.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of is already in our list. That factor is However, the other factor of is not listed in step 2. We add a second factor of to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus,
is the least common denominator.
In Example 7, if you were to multiply the denominators as given, you would get a common denominator, but not the LCD. While any common denominator can be used to add or subtract rational expressions, using the LCD greatly reduces the amount of work involved.
Finding the least common denominator for two (or more) rational expressions is the first step needed to add or subtract the expressions.
Adding and Subtracting Rational Expressions That Have Different Denominators
Find the LCD of the rational expressions.
Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
Add or subtract numerators, placing the resulting expression over the LCD.
If possible, simplify the resulting rational expression.
Example 8 Adding and Subtracting Rational Expressions with Different Denominators
Add or subtract:
.
Solution
Note that each of these denominators has only one term.
The second rational expression in the sum, , is undefined for , so .
Step 1 FIND THE LEAST COMMON DENOMINATOR. The denominators are 3 and x, which have no common factors. Their product, 3x, is the LCD.
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of 3x. We do so by multiplying the numerator and denominator of by x and multiplying the numerator and denominator of by 3.
Because and , we are not changing the value of either rational expression. Now we are ready to perform the addition.
Step 3 ADD NUMERATORS, PLACING THIS SUM OVER THE LCD.
Step 4 Simplify, if necessary. Because the numerator is prime, no further simplification is possible.
Both rational expressions in the subtraction are undefined for so .
Step 1 Find the least common denominator. The denominators are 6x and . Begin by listing the factors of 6x:
The factors of are 3, 3, x, and x. The list of factors of 6x only contains one factor of 3 and one factor of x, so we add one more 3 and one more x to the list:
The LCD is the product of the factors in this list. The LCD is .
Step 2 Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of . We do so by multiplying the numerator and denominator of by 3x and multiplying the numerator and denominator of by 2.
Example 9 Subtracting Rational Expressions with Different Denominators
Subtract: .
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. In Example 6, we found that the LCD for these rational expressions is .
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of . We do so by multiplying both the numerator and the denominator of each rational expression by any factor needed to convert the expression’s denominator into the LCD.
Example 10 Adding Rational Expressions with Different Denominators
Add:
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. Start by factoring the denominators.
The factors of the first denominator are and The only factor from the second denominator that is not listed is Thus, the least common denominator is
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of We do so by multiplying both the numerator and the denominator of each rational expression by any factor(s) needed to convert the expression’s denominator into the LCD.
Objective 5: Add and subtract rational expressions
Adding and Subtracting Rational Expressions with the Same Denominator
Objective 5Add and subtract rational expressions.
Watch Video
We add or subtract rational expressions with the same denominator by (1) adding or subtracting the numerators, (2) placing this result over the common denominator, and (3) simplifying, if possible.
Example 5 Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Different Denominators
We can gain insight into adding rational expressions with different denominators by looking closely at what we do when adding fractions with different denominators. For example, suppose that we want to add and We must first write the fractions with the same denominator. We look for the smallest number that contains both 2 and 3 as factors. This number, 6, is then used as the least common denominator, or LCD.
The least common denominator, or LCD, of several rational expressions is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
Finding the Least Common Denominator
Factor each denominator completely.
List the factors of the first denominator.
Add to the list in step 2 any factors of the second denominator that do not appear in the list.
Form the product of each different factor from the list in step 3. This product is the least common denominator.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of the second denominator is already in our list. That factor is 1. However, the other factor, , is not listed in step 2. We add to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus
In Example 6 and Check Point 6, the denominators do not have any common factors other than 1 and . Thus, the LCD is simply the product of the denominators. However, in Example 7, the denominators both have a factor of .
Example 7 Finding the Least Common Denominator
Find the LCD of
Solution
Step 1 FACTOR EACH DENOMINATOR COMPLETELY.
Step 2 LIST THE FACTORS OF THE FIRST DENOMINATOR.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of is already in our list. That factor is However, the other factor of is not listed in step 2. We add a second factor of to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus,
is the least common denominator.
In Example 7, if you were to multiply the denominators as given, you would get a common denominator, but not the LCD. While any common denominator can be used to add or subtract rational expressions, using the LCD greatly reduces the amount of work involved.
Finding the least common denominator for two (or more) rational expressions is the first step needed to add or subtract the expressions.
Adding and Subtracting Rational Expressions That Have Different Denominators
Find the LCD of the rational expressions.
Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
Add or subtract numerators, placing the resulting expression over the LCD.
If possible, simplify the resulting rational expression.
Example 8 Adding and Subtracting Rational Expressions with Different Denominators
Add or subtract:
.
Solution
Note that each of these denominators has only one term.
The second rational expression in the sum, , is undefined for , so .
Step 1 FIND THE LEAST COMMON DENOMINATOR. The denominators are 3 and x, which have no common factors. Their product, 3x, is the LCD.
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of 3x. We do so by multiplying the numerator and denominator of by x and multiplying the numerator and denominator of by 3.
Because and , we are not changing the value of either rational expression. Now we are ready to perform the addition.
Step 3 ADD NUMERATORS, PLACING THIS SUM OVER THE LCD.
Step 4 Simplify, if necessary. Because the numerator is prime, no further simplification is possible.
Both rational expressions in the subtraction are undefined for so .
Step 1 Find the least common denominator. The denominators are 6x and . Begin by listing the factors of 6x:
The factors of are 3, 3, x, and x. The list of factors of 6x only contains one factor of 3 and one factor of x, so we add one more 3 and one more x to the list:
The LCD is the product of the factors in this list. The LCD is .
Step 2 Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of . We do so by multiplying the numerator and denominator of by 3x and multiplying the numerator and denominator of by 2.
Example 9 Subtracting Rational Expressions with Different Denominators
Subtract: .
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. In Example 6, we found that the LCD for these rational expressions is .
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of . We do so by multiplying both the numerator and the denominator of each rational expression by any factor needed to convert the expression’s denominator into the LCD.
Example 10 Adding Rational Expressions with Different Denominators
Add:
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. Start by factoring the denominators.
The factors of the first denominator are and The only factor from the second denominator that is not listed is Thus, the least common denominator is
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of We do so by multiplying both the numerator and the denominator of each rational expression by any factor(s) needed to convert the expression’s denominator into the LCD.
Objective 5: Add and subtract rational expressions
Adding and Subtracting Rational Expressions with the Same Denominator
Objective 5Add and subtract rational expressions.
Watch Video
We add or subtract rational expressions with the same denominator by (1) adding or subtracting the numerators, (2) placing this result over the common denominator, and (3) simplifying, if possible.
Example 5 Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Different Denominators
We can gain insight into adding rational expressions with different denominators by looking closely at what we do when adding fractions with different denominators. For example, suppose that we want to add and We must first write the fractions with the same denominator. We look for the smallest number that contains both 2 and 3 as factors. This number, 6, is then used as the least common denominator, or LCD.
The least common denominator, or LCD, of several rational expressions is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
Finding the Least Common Denominator
Factor each denominator completely.
List the factors of the first denominator.
Add to the list in step 2 any factors of the second denominator that do not appear in the list.
Form the product of each different factor from the list in step 3. This product is the least common denominator.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of the second denominator is already in our list. That factor is 1. However, the other factor, , is not listed in step 2. We add to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus
In Example 6 and Check Point 6, the denominators do not have any common factors other than 1 and . Thus, the LCD is simply the product of the denominators. However, in Example 7, the denominators both have a factor of .
Example 7 Finding the Least Common Denominator
Find the LCD of
Solution
Step 1 FACTOR EACH DENOMINATOR COMPLETELY.
Step 2 LIST THE FACTORS OF THE FIRST DENOMINATOR.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of is already in our list. That factor is However, the other factor of is not listed in step 2. We add a second factor of to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus,
is the least common denominator.
In Example 7, if you were to multiply the denominators as given, you would get a common denominator, but not the LCD. While any common denominator can be used to add or subtract rational expressions, using the LCD greatly reduces the amount of work involved.
Finding the least common denominator for two (or more) rational expressions is the first step needed to add or subtract the expressions.
Adding and Subtracting Rational Expressions That Have Different Denominators
Find the LCD of the rational expressions.
Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
Add or subtract numerators, placing the resulting expression over the LCD.
If possible, simplify the resulting rational expression.
Example 8 Adding and Subtracting Rational Expressions with Different Denominators
Add or subtract:
.
Solution
Note that each of these denominators has only one term.
The second rational expression in the sum, , is undefined for , so .
Step 1 FIND THE LEAST COMMON DENOMINATOR. The denominators are 3 and x, which have no common factors. Their product, 3x, is the LCD.
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of 3x. We do so by multiplying the numerator and denominator of by x and multiplying the numerator and denominator of by 3.
Because and , we are not changing the value of either rational expression. Now we are ready to perform the addition.
Step 3 ADD NUMERATORS, PLACING THIS SUM OVER THE LCD.
Step 4 Simplify, if necessary. Because the numerator is prime, no further simplification is possible.
Both rational expressions in the subtraction are undefined for so .
Step 1 Find the least common denominator. The denominators are 6x and . Begin by listing the factors of 6x:
The factors of are 3, 3, x, and x. The list of factors of 6x only contains one factor of 3 and one factor of x, so we add one more 3 and one more x to the list:
The LCD is the product of the factors in this list. The LCD is .
Step 2 Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of . We do so by multiplying the numerator and denominator of by 3x and multiplying the numerator and denominator of by 2.
Example 9 Subtracting Rational Expressions with Different Denominators
Subtract: .
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. In Example 6, we found that the LCD for these rational expressions is .
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of . We do so by multiplying both the numerator and the denominator of each rational expression by any factor needed to convert the expression’s denominator into the LCD.
Example 10 Adding Rational Expressions with Different Denominators
Add:
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. Start by factoring the denominators.
The factors of the first denominator are and The only factor from the second denominator that is not listed is Thus, the least common denominator is
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of We do so by multiplying both the numerator and the denominator of each rational expression by any factor(s) needed to convert the expression’s denominator into the LCD.
Objective 5: Add and subtract rational expressions
Adding and Subtracting Rational Expressions with the Same Denominator
Objective 5Add and subtract rational expressions.
Watch Video
We add or subtract rational expressions with the same denominator by (1) adding or subtracting the numerators, (2) placing this result over the common denominator, and (3) simplifying, if possible.
Example 5 Subtracting Rational Expressions with the Same Denominator
Adding and Subtracting Rational Expressions with Different Denominators
We can gain insight into adding rational expressions with different denominators by looking closely at what we do when adding fractions with different denominators. For example, suppose that we want to add and We must first write the fractions with the same denominator. We look for the smallest number that contains both 2 and 3 as factors. This number, 6, is then used as the least common denominator, or LCD.
The least common denominator, or LCD, of several rational expressions is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator.
Finding the Least Common Denominator
Factor each denominator completely.
List the factors of the first denominator.
Add to the list in step 2 any factors of the second denominator that do not appear in the list.
Form the product of each different factor from the list in step 3. This product is the least common denominator.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of the second denominator is already in our list. That factor is 1. However, the other factor, , is not listed in step 2. We add to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus
In Example 6 and Check Point 6, the denominators do not have any common factors other than 1 and . Thus, the LCD is simply the product of the denominators. However, in Example 7, the denominators both have a factor of .
Example 7 Finding the Least Common Denominator
Find the LCD of
Solution
Step 1 FACTOR EACH DENOMINATOR COMPLETELY.
Step 2 LIST THE FACTORS OF THE FIRST DENOMINATOR.
Step 3 ADD ANY UNLISTED FACTORS FROM THE SECOND DENOMINATOR. One factor of is already in our list. That factor is However, the other factor of is not listed in step 2. We add a second factor of to the list. We have
Step 4 THE LEAST COMMON DENOMINATOR IS THE PRODUCT OF ALL FACTORS IN THE FINAL LIST. Thus,
is the least common denominator.
In Example 7, if you were to multiply the denominators as given, you would get a common denominator, but not the LCD. While any common denominator can be used to add or subtract rational expressions, using the LCD greatly reduces the amount of work involved.
Finding the least common denominator for two (or more) rational expressions is the first step needed to add or subtract the expressions.
Adding and Subtracting Rational Expressions That Have Different Denominators
Find the LCD of the rational expressions.
Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
Add or subtract numerators, placing the resulting expression over the LCD.
If possible, simplify the resulting rational expression.
Example 8 Adding and Subtracting Rational Expressions with Different Denominators
Add or subtract:
.
Solution
Note that each of these denominators has only one term.
The second rational expression in the sum, , is undefined for , so .
Step 1 FIND THE LEAST COMMON DENOMINATOR. The denominators are 3 and x, which have no common factors. Their product, 3x, is the LCD.
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of 3x. We do so by multiplying the numerator and denominator of by x and multiplying the numerator and denominator of by 3.
Because and , we are not changing the value of either rational expression. Now we are ready to perform the addition.
Step 3 ADD NUMERATORS, PLACING THIS SUM OVER THE LCD.
Step 4 Simplify, if necessary. Because the numerator is prime, no further simplification is possible.
Both rational expressions in the subtraction are undefined for so .
Step 1 Find the least common denominator. The denominators are 6x and . Begin by listing the factors of 6x:
The factors of are 3, 3, x, and x. The list of factors of 6x only contains one factor of 3 and one factor of x, so we add one more 3 and one more x to the list:
The LCD is the product of the factors in this list. The LCD is .
Step 2 Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of . We do so by multiplying the numerator and denominator of by 3x and multiplying the numerator and denominator of by 2.
Example 9 Subtracting Rational Expressions with Different Denominators
Subtract: .
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. In Example 6, we found that the LCD for these rational expressions is .
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of . We do so by multiplying both the numerator and the denominator of each rational expression by any factor needed to convert the expression’s denominator into the LCD.
Example 10 Adding Rational Expressions with Different Denominators
Add:
Solution
Step 1 FIND THE LEAST COMMON DENOMINATOR. Start by factoring the denominators.
The factors of the first denominator are and The only factor from the second denominator that is not listed is Thus, the least common denominator is
Step 2 WRITE EQUIVALENT EXPRESSIONS WITH THE LCD AS DENOMINATORS. We must rewrite each rational expression with a denominator of We do so by multiplying both the numerator and the denominator of each rational expression by any factor(s) needed to convert the expression’s denominator into the LCD.
Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more rational expressions. Here are two examples of such expressions:
One method for simplifying a complex rational expression is to combine its numerator into a single expression and combine its denominator into a single expression. Then perform the division by inverting the denominator and multiplying.
Example 11 Simplifying a Complex Rational Expression
Simplify:
Solution
Step 1 ADD TO GET A SINGLE RATIONAL EXPRESSION IN THE NUMERATOR.
A second method for simplifying a complex rational expression is to find the least common denominator of all the rational expressions in its numerator and denominator. Then multiply each term in its numerator and denominator by this least common denominator. Because we are multiplying by a form of 1, we will obtain an equivalent expression that does not contain fractions in its numerator or denominator. Here we use this method to simplify the complex rational expression in Example 11.
Example 12 Simplifying a Complex Rational Expression
Simplify:
Solution
We will use the method of multiplying each of the three terms, and h, by the least common denominator. The least common denominator is
Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more rational expressions. Here are two examples of such expressions:
One method for simplifying a complex rational expression is to combine its numerator into a single expression and combine its denominator into a single expression. Then perform the division by inverting the denominator and multiplying.
Example 11 Simplifying a Complex Rational Expression
Simplify:
Solution
Step 1 ADD TO GET A SINGLE RATIONAL EXPRESSION IN THE NUMERATOR.
A second method for simplifying a complex rational expression is to find the least common denominator of all the rational expressions in its numerator and denominator. Then multiply each term in its numerator and denominator by this least common denominator. Because we are multiplying by a form of 1, we will obtain an equivalent expression that does not contain fractions in its numerator or denominator. Here we use this method to simplify the complex rational expression in Example 11.
Example 12 Simplifying a Complex Rational Expression
Simplify:
Solution
We will use the method of multiplying each of the three terms, and h, by the least common denominator. The least common denominator is
Objective 7: Simplify fractional expressions that occur in calculus
Fractional Expressions in Calculus
Objective 7Simplify fractional expressions that occur in calculus.
Watch Video
Fractional expressions containing radicals occur frequently in calculus. Because of the radicals, these expressions are not rational expressions. However, they can often be simplified using the procedure for simplifying complex rational expressions.
Example 13 Simplifying a Fractional Expression Containing Radicals
Another fractional expression that you will encounter in calculus is
Can you see that this expression is not defined if However, in calculus, you will ask the following question:
What happens to the expression as h takes on values that get closer and closer to 0, such as and so on?
The question is answered by first rationalizing the numerator. This process involves rewriting the fractional expression as an equivalent expression in which the numerator no longer contains any radicals. To rationalize a numerator, multiply by 1 to eliminate the radicals in the numerator. Multiply the numerator and the denominator by the conjugate of the numerator.
Example 14 Rationalizing a Numerator
Rationalize the numerator:
Solution
The conjugate of the numerator is If we multiply the numerator and denominator by the simplified numerator will not contain a radical. Therefore, we multiply by 1, choosing for 1.
What happens to as h gets closer and closer to 0? In Example 14, we showed that
As h gets closer to 0, the expression on the right gets closer to or Thus, the fractional expression approaches as h gets closer to 0.
Fill in each blank so that the resulting statement is true.
C1. A rational expression is the quotient of two _________.
C2. The set of real numbers for which a rational expression is defined is the _________ of the expression. We must exclude all numbers from this set that make the denominator of the rational expression ______.
C3. We simplify a rational expression by _________ the numerator and the denominator completely. Then we divide the numerator and the denominator by any ______________.
The factors of the first denominator are _____________.
The factors of the second denominator are ________.
The LCD is ____________.
C8. An equivalent expression for with a denominator of can be obtained by multiplying the numerator and denominator by ______.
C9. A rational expression whose numerator or denominator or both contain rational expressions is called a/an ________ rational expression or a/an ________ fraction.
by multiplying the numerator and the denominator by ______.
C12. We can rationalize the numerator of by multiplying the numerator and the denominator by ______.
P.6: Exercise Set
P.6 Exercise Set
Practice Exercises
In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression.
1.
2.
3.
4.
5.
6.
In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
7.
8.
9.
10.
11.
12.
13.
14.
In Exercises 15–32, multiply or divide as indicated.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–68, add or subtract as indicated.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
In Exercises 69–82, simplify each complex rational expression.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
Exercises 83–88 contain fractional expressions that occur frequently in calculus. Simplify each expression.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, rationalize the numerator.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–100, perform the indicated operations. Simplify the result, if possible.
93.
94.
95.
96.
97.
98.
99.
100.
Application Exercises
101. The rational expression
describes the cost, in millions of dollars, to inoculate x percent of the population against a particular strain of flu.
Evaluate the expression for and Describe the meaning of each evaluation in terms of percentage inoculated and cost.
For what value of x is the expression undefined?
What happens to the cost as x approaches 100%? How can you interpret this observation?
102. The average rate on a round-trip commute having a one-way distance d is given by the complex rational expression
in which and are the average rates on the outgoing and return trips, respectively. Simplify the expression. Then find your average rate if you drive to campus averaging 40 miles per hour and return home on the same route averaging 30 miles per hour. Explain why the answer is not 35 miles per hour.
103. The bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderately active lifestyles. (Moderately active means a lifestyle that includes physical activity equivalent to walking 1.5 to 3 miles per day at 3 to 4 miles per hour, in addition to the light physical activity associated with typical day-to-day life.)
describes the number of calories needed per day, W, by women in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by women between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much?
The mathematical model
describes the number of calories needed per day, M, by men in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by men between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much?
Write a simplified rational expression that describes the ratio of the number of calories needed per day by women in age group x to the number of calories needed per day by men in age group x for people with moderately active lifestyles.
104. If three resistors with resistances , , and are connected in parallel, their combined resistance is given by the expression
Simplify the complex rational expression. Then find the combined resistance when is 4 ohms, is 8 ohms, and is 12 ohms.
In Exercises 105–106, express the perimeter of each rectangle as a single rational expression.
105.
106.
Explaining the Concepts
107. What is a rational expression?
108. Explain how to determine which numbers must be excluded from the domain of a rational expression.
109. Explain how to simplify a rational expression.
110. Explain how to multiply rational expressions.
111. Explain how to divide rational expressions.
112. Explain how to add or subtract rational expressions with the same denominators.
113. Explain how to add rational expressions with different denominators. Use in your explanation.
114. Explain how to find the least common denominator for denominators of and
115. Describe two ways to simplify
Explain the error in Exercises 116–118. Then rewrite the right side of the equation to correct the error that now exists.
116.
117.
118.
Critical Thinking Exercises
Make Sense? In Exercises 119–122, determine whether each statement makes sense or does not make sense, and explain your reasoning.
119. I evaluated for and obtained 0.
120. The rational expressions
can both be simplified by dividing each numerator and each denominator by 7.
121. When performing the division
I began by dividing the numerator and the denominator by the common factor,
122. I subtracted from and obtained a constant.
In Exercises 123–126, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
123.
124. The expression simplifies to the consecutive integer that follows
125.
126.
In Exercises 127–129, perform the indicated operations.
127.
128.
129.
130. In one short sentence, five words or less, explain what
does to each number x.
Preview Exercises
Exercises 131–133 will help you prepare for the material covered in the next section.
131. If 6 is substituted for x in the equation
is the resulting statement true or false?
132. Multiply and simplify:
133. Evaluate
for .
P.6: Exercise Set
P.6 Exercise Set
Practice Exercises
In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression.
1.
2.
3.
4.
5.
6.
In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
7.
8.
9.
10.
11.
12.
13.
14.
In Exercises 15–32, multiply or divide as indicated.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–68, add or subtract as indicated.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
In Exercises 69–82, simplify each complex rational expression.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
Exercises 83–88 contain fractional expressions that occur frequently in calculus. Simplify each expression.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, rationalize the numerator.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–100, perform the indicated operations. Simplify the result, if possible.
93.
94.
95.
96.
97.
98.
99.
100.
Application Exercises
101. The rational expression
describes the cost, in millions of dollars, to inoculate x percent of the population against a particular strain of flu.
Evaluate the expression for and Describe the meaning of each evaluation in terms of percentage inoculated and cost.
For what value of x is the expression undefined?
What happens to the cost as x approaches 100%? How can you interpret this observation?
102. The average rate on a round-trip commute having a one-way distance d is given by the complex rational expression
in which and are the average rates on the outgoing and return trips, respectively. Simplify the expression. Then find your average rate if you drive to campus averaging 40 miles per hour and return home on the same route averaging 30 miles per hour. Explain why the answer is not 35 miles per hour.
103. The bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderately active lifestyles. (Moderately active means a lifestyle that includes physical activity equivalent to walking 1.5 to 3 miles per day at 3 to 4 miles per hour, in addition to the light physical activity associated with typical day-to-day life.)
describes the number of calories needed per day, W, by women in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by women between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much?
The mathematical model
describes the number of calories needed per day, M, by men in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by men between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much?
Write a simplified rational expression that describes the ratio of the number of calories needed per day by women in age group x to the number of calories needed per day by men in age group x for people with moderately active lifestyles.
104. If three resistors with resistances , , and are connected in parallel, their combined resistance is given by the expression
Simplify the complex rational expression. Then find the combined resistance when is 4 ohms, is 8 ohms, and is 12 ohms.
In Exercises 105–106, express the perimeter of each rectangle as a single rational expression.
105.
106.
Explaining the Concepts
107. What is a rational expression?
108. Explain how to determine which numbers must be excluded from the domain of a rational expression.
109. Explain how to simplify a rational expression.
110. Explain how to multiply rational expressions.
111. Explain how to divide rational expressions.
112. Explain how to add or subtract rational expressions with the same denominators.
113. Explain how to add rational expressions with different denominators. Use in your explanation.
114. Explain how to find the least common denominator for denominators of and
115. Describe two ways to simplify
Explain the error in Exercises 116–118. Then rewrite the right side of the equation to correct the error that now exists.
116.
117.
118.
Critical Thinking Exercises
Make Sense? In Exercises 119–122, determine whether each statement makes sense or does not make sense, and explain your reasoning.
119. I evaluated for and obtained 0.
120. The rational expressions
can both be simplified by dividing each numerator and each denominator by 7.
121. When performing the division
I began by dividing the numerator and the denominator by the common factor,
122. I subtracted from and obtained a constant.
In Exercises 123–126, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
123.
124. The expression simplifies to the consecutive integer that follows
125.
126.
In Exercises 127–129, perform the indicated operations.
127.
128.
129.
130. In one short sentence, five words or less, explain what
does to each number x.
Preview Exercises
Exercises 131–133 will help you prepare for the material covered in the next section.
131. If 6 is substituted for x in the equation
is the resulting statement true or false?
132. Multiply and simplify:
133. Evaluate
for .
P.6: Exercise Set
P.6 Exercise Set
Practice Exercises
In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression.
1.
2.
3.
4.
5.
6.
In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
7.
8.
9.
10.
11.
12.
13.
14.
In Exercises 15–32, multiply or divide as indicated.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–68, add or subtract as indicated.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
In Exercises 69–82, simplify each complex rational expression.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
Exercises 83–88 contain fractional expressions that occur frequently in calculus. Simplify each expression.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, rationalize the numerator.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–100, perform the indicated operations. Simplify the result, if possible.
93.
94.
95.
96.
97.
98.
99.
100.
Application Exercises
101. The rational expression
describes the cost, in millions of dollars, to inoculate x percent of the population against a particular strain of flu.
Evaluate the expression for and Describe the meaning of each evaluation in terms of percentage inoculated and cost.
For what value of x is the expression undefined?
What happens to the cost as x approaches 100%? How can you interpret this observation?
102. The average rate on a round-trip commute having a one-way distance d is given by the complex rational expression
in which and are the average rates on the outgoing and return trips, respectively. Simplify the expression. Then find your average rate if you drive to campus averaging 40 miles per hour and return home on the same route averaging 30 miles per hour. Explain why the answer is not 35 miles per hour.
103. The bar graph shows the estimated number of calories per day needed to maintain energy balance for various gender and age groups for moderately active lifestyles. (Moderately active means a lifestyle that includes physical activity equivalent to walking 1.5 to 3 miles per day at 3 to 4 miles per hour, in addition to the light physical activity associated with typical day-to-day life.)
describes the number of calories needed per day, W, by women in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by women between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much?
The mathematical model
describes the number of calories needed per day, M, by men in age group x with moderately active lifestyles. According to the model, how many calories per day are needed by men between the ages of 19 and 30, inclusive, with this lifestyle? Does this underestimate or overestimate the number shown by the graph? By how much?
Write a simplified rational expression that describes the ratio of the number of calories needed per day by women in age group x to the number of calories needed per day by men in age group x for people with moderately active lifestyles.
104. If three resistors with resistances , , and are connected in parallel, their combined resistance is given by the expression
Simplify the complex rational expression. Then find the combined resistance when is 4 ohms, is 8 ohms, and is 12 ohms.
In Exercises 105–106, express the perimeter of each rectangle as a single rational expression.
105.
106.
Explaining the Concepts
107. What is a rational expression?
108. Explain how to determine which numbers must be excluded from the domain of a rational expression.
109. Explain how to simplify a rational expression.
110. Explain how to multiply rational expressions.
111. Explain how to divide rational expressions.
112. Explain how to add or subtract rational expressions with the same denominators.
113. Explain how to add rational expressions with different denominators. Use in your explanation.
114. Explain how to find the least common denominator for denominators of and
115. Describe two ways to simplify
Explain the error in Exercises 116–118. Then rewrite the right side of the equation to correct the error that now exists.
116.
117.
118.
Critical Thinking Exercises
Make Sense? In Exercises 119–122, determine whether each statement makes sense or does not make sense, and explain your reasoning.
119. I evaluated for and obtained 0.
120. The rational expressions
can both be simplified by dividing each numerator and each denominator by 7.
121. When performing the division
I began by dividing the numerator and the denominator by the common factor,
122. I subtracted from and obtained a constant.
In Exercises 123–126, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
123.
124. The expression simplifies to the consecutive integer that follows
125.
126.
In Exercises 127–129, perform the indicated operations.
127.
128.
129.
130. In one short sentence, five words or less, explain what
does to each number x.
Preview Exercises
Exercises 131–133 will help you prepare for the material covered in the next section.
I’m very well acquainted, too, with matters mathematical, I understand equations, both simple and quadratical. About binomial theorem I’m teeming with a lot of news, With many cheerful facts about the square of the hypotenuse.
—Gilbert and Sullivan, The Pirates of Penzance
Equations quadratical? Cheerful news about the square of the hypotenuse? You’ve come to the right place. In this section, we will review how to solve a variety of equations, including linear equations, quadratic equations, and radical equations. (Yes, it’s quadratic and not quadratical, despite the latter’s rhyme with mathematical.) In the next section, we will look at applications of quadratic equations, introducing (cheerfully, of course) the Pythagorean Theorem and the square of the hypotenuse.
Objective 1: Solve linear equations in one variable
Solving Linear Equations in One Variable
Objective 1Solve linear equations in one variable.
Watch Video
We begin with a general definition of a linear equation in one variable.
Definition of a Linear Equation
A linear equation in one variable x is an equation that can be written in the form
where a and are real numbers, and
An example of a linear equation in one variable is
Solving an equation in x involves determining all values of x that result in a true statement when substituted into the equation. Such values are solutions, or roots, of the equation. For example, substitute for x in We obtain
This simplifies to the true statement Thus, is a solution of the equation We also say that satisfies the equation because when we substitute for x, a true statement results. The set of all such solutions is called the equation’s solution set. For example, the solution set of the equation is because is the equation’s only solution.
Two or more equations that have the same solution set are called equivalent equations. For example, the equations
are equivalent equations because the solution set for each is To solve a linear equation in x, we transform the equation into an equivalent equation one or more times. Our final equivalent equation should be of the form
The solution set of this equation is the set consisting of the number.
To generate equivalent equations, we will use the principles in the box.
Generating Equivalent Equations
An equation can be transformed into an equivalent equation by one or more of the following operations:
If you look closely at the equations in the box, you will notice that we have solved the equation The final equation, with x isolated on the left side, shows that is the solution set. The idea in solving a linear equation is to get the variable by itself on one side of the equal sign and a number by itself on the other side.
Here is a step-by-step procedure for solving a linear equation in one variable. Not all of these steps are necessary to solve every equation.
Solving a Linear Equation
Simplify the algebraic expression on each side by removing grouping symbols and combining like terms.
Collect all the variable terms on one side and all the numbers, or constant terms, on the other side.
Isolate the variable and solve.
Check the proposed solution in the original equation.
Example 1 Solving a Linear Equation
Solve and check:
Solution
Step 1 SIMPLIFY THE ALGEBRAIC EXPRESSION ON EACH SIDE.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left by adding to both sides. We will collect the numbers on the right by adding 23 to both sides.
Step 3. ISOLATE THE VARIABLE AND SOLVE. We isolate the variable, x, by dividing both sides of by 5.
Step 4. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL EQUATION. Substitute 6 for x in the original equation.
The true statement verifies that the solution set is
Objective 1: Solve linear equations in one variable
Solving Linear Equations in One Variable
Objective 1Solve linear equations in one variable.
Watch Video
We begin with a general definition of a linear equation in one variable.
Definition of a Linear Equation
A linear equation in one variable x is an equation that can be written in the form
where a and are real numbers, and
An example of a linear equation in one variable is
Solving an equation in x involves determining all values of x that result in a true statement when substituted into the equation. Such values are solutions, or roots, of the equation. For example, substitute for x in We obtain
This simplifies to the true statement Thus, is a solution of the equation We also say that satisfies the equation because when we substitute for x, a true statement results. The set of all such solutions is called the equation’s solution set. For example, the solution set of the equation is because is the equation’s only solution.
Two or more equations that have the same solution set are called equivalent equations. For example, the equations
are equivalent equations because the solution set for each is To solve a linear equation in x, we transform the equation into an equivalent equation one or more times. Our final equivalent equation should be of the form
The solution set of this equation is the set consisting of the number.
To generate equivalent equations, we will use the principles in the box.
Generating Equivalent Equations
An equation can be transformed into an equivalent equation by one or more of the following operations:
If you look closely at the equations in the box, you will notice that we have solved the equation The final equation, with x isolated on the left side, shows that is the solution set. The idea in solving a linear equation is to get the variable by itself on one side of the equal sign and a number by itself on the other side.
Here is a step-by-step procedure for solving a linear equation in one variable. Not all of these steps are necessary to solve every equation.
Solving a Linear Equation
Simplify the algebraic expression on each side by removing grouping symbols and combining like terms.
Collect all the variable terms on one side and all the numbers, or constant terms, on the other side.
Isolate the variable and solve.
Check the proposed solution in the original equation.
Example 1 Solving a Linear Equation
Solve and check:
Solution
Step 1 SIMPLIFY THE ALGEBRAIC EXPRESSION ON EACH SIDE.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left by adding to both sides. We will collect the numbers on the right by adding 23 to both sides.
Step 3. ISOLATE THE VARIABLE AND SOLVE. We isolate the variable, x, by dividing both sides of by 5.
Step 4. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL EQUATION. Substitute 6 for x in the original equation.
The true statement verifies that the solution set is
Objective 2: Solve linear equations containing fractions
Linear Equations with Fractions
Objective 2Solve linear equations containing fractions.
Watch Video
Equations are easier to solve when they do not contain fractions. How do we remove fractions from an equation? We begin by multiplying both sides of the equation by the least common denominator of any fractions in the equation. The least common denominator is the smallest number that all denominators will divide into. Multiplying every term on both sides of the equation by the least common denominator will eliminate the fractions in the equation. Example 2 shows how we “clear an equation of fractions.”
Example 2 Solving a Linear Equation Involving Fractions
Solve and check:
Solution
The fractional terms have denominators of 4 and 3. The smallest number that is divisible by 4 and 3 is 12. We begin by multiplying both sides of the equation by 12, the least common denominator.
Objective 3: Solve rational equations with variables in the denominators
Rational Equations
Objective 3Solve rational equations with variables in the denominators.
Watch Video
A rational equation is an equation containing one or more rational expressions. In Example 2, we solved a rational equation with constants in the denominators. This rational equation was a linear equation. Now, let’s consider a rational equation such as
Can you see how this rational equation differs from the rational equation that we solved earlier? The variable appears in the denominators. Although this rational equation is not a linear equation, the solution procedure still involves multiplying each side by the least common denominator. However, we must avoid any values of the variable that make a denominator zero.
Example 3 Solving a Rational Equation
Solve:
Solution
To identify values of x that make denominators zero, let’s factor the denominator on the right. This factorization is also necessary in identifying the least common denominator.
The proposed solution, 1, is not a solution because of the restriction that There is no solution to this equation. The solution set for this equation contains no elements. The solution set is Ø, the empty set.
Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation. It does not mean obtaining a numerical value for that variable.
To solve a formula for one of its variables, treat that variable as if it were the only variable in the equation. Think of the other variables as if they were numbers.
Example 5 Solving a Formula for a Variable
If you wear glasses, did you know that each lens has a measurement called its focal length, f? When an object is in focus, its distance from the lens, p, and the distance from the lens to your retina, q, satisfy the formula
To collect terms with p on one side of subtract pf from both sides. Then factor p from the two resulting terms on the right to convert two occurrences of p into one.
We have seen that the absolute value of x, denoted describes the distance of x from zero on a number line. Now consider an absolute value equation, such as
This means that we must determine real numbers whose distance from the origin on a number line is 2. Figure P.13 shows that there are two numbers such that namely, 2 and We write or This observation can be generalized as follows:
The absolute value of a number is never negative. Thus, if u is an algebraic expression and c is a negative number, then has no solution. For example, the equation has no solution because cannot be negative. The solution set is Ø, the empty set.
The absolute value of 0 is 0. Thus, if u is an algebraic expression and the solution is found by solving For example, the solution of is obtained by solving The solution is 2 and the solution set is
Objective 6: Solve quadratic equations by factoring
Quadratic Equations and Factoring
Objective 6Solve quadratic equations by factoring.
Watch Video
Linear equations are first-degree polynomial equations of the form Quadratic equations are second-degree polynomial equations and contain an additional term involving the square of the variable.
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written in the standard form
where and c are real numbers, with A quadratic equation in x is also called a second-degree polynomial equation inx.
Here are examples of quadratic equations in standard form:
Some quadratic equations, including the two shown above, can be solved by factoring and using the zero-product principle.
The Zero-Product Principle
If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.
The zero-product principle can be applied only when a quadratic equation is in standard form, with zero on one side of the equation.
Solving a Quadratic Equation by Factoring
If necessary, rewrite the equation in the standard form moving all nonzero terms to one side, thereby obtaining zero on the other side.
Factor completely.
Apply the zero-product principle, setting each factor containing a variable equal to zero.
Solve the equations in step 3.
Check the solutions in the original equation.
Example 7 Solving Quadratic Equations by Factoring
Solve by factoring:
Solution
We begin with
Step 1 MOVE ALL NONZERO TERMS TO ONE SIDE AND OBTAIN ZERO ON THE OTHER SIDE. All nonzero terms are already on the left and zero is on the other side, so we can skip this step.
Step 2 FACTOR. We factor out from the two terms on the left side.
Steps 3 and 4 SET EACH FACTOR EQUAL TO ZERO AND SOLVE THE RESULTING EQUATIONS. We apply the zero-product principle to
Step 5 CHECK THE SOLUTIONS IN THE ORIGINAL EQUATION.
Check 0:
Check :
The solution set is
Next, we solve
Step 1 MOVE ALL NONZERO TERMS TO ONE SIDE AND OBTAIN ZERO ON THE OTHER SIDE. Subtract 4 from both sides and write the equation in standard form.
Step 2 FACTOR.
Steps 3 and 4 SET EACH FACTOR EQUAL TO ZERO AND SOLVE THE RESULTING EQUATIONS.
Step 5 CHECK THE SOLUTIONS IN THE ORIGINAL EQUATION.
Objective 6: Solve quadratic equations by factoring
Quadratic Equations and Factoring
Objective 6Solve quadratic equations by factoring.
Watch Video
Linear equations are first-degree polynomial equations of the form Quadratic equations are second-degree polynomial equations and contain an additional term involving the square of the variable.
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written in the standard form
where and c are real numbers, with A quadratic equation in x is also called a second-degree polynomial equation inx.
Here are examples of quadratic equations in standard form:
Some quadratic equations, including the two shown above, can be solved by factoring and using the zero-product principle.
The Zero-Product Principle
If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.
The zero-product principle can be applied only when a quadratic equation is in standard form, with zero on one side of the equation.
Solving a Quadratic Equation by Factoring
If necessary, rewrite the equation in the standard form moving all nonzero terms to one side, thereby obtaining zero on the other side.
Factor completely.
Apply the zero-product principle, setting each factor containing a variable equal to zero.
Solve the equations in step 3.
Check the solutions in the original equation.
Example 7 Solving Quadratic Equations by Factoring
Solve by factoring:
Solution
We begin with
Step 1 MOVE ALL NONZERO TERMS TO ONE SIDE AND OBTAIN ZERO ON THE OTHER SIDE. All nonzero terms are already on the left and zero is on the other side, so we can skip this step.
Step 2 FACTOR. We factor out from the two terms on the left side.
Steps 3 and 4 SET EACH FACTOR EQUAL TO ZERO AND SOLVE THE RESULTING EQUATIONS. We apply the zero-product principle to
Step 5 CHECK THE SOLUTIONS IN THE ORIGINAL EQUATION.
Check 0:
Check :
The solution set is
Next, we solve
Step 1 MOVE ALL NONZERO TERMS TO ONE SIDE AND OBTAIN ZERO ON THE OTHER SIDE. Subtract 4 from both sides and write the equation in standard form.
Step 2 FACTOR.
Steps 3 and 4 SET EACH FACTOR EQUAL TO ZERO AND SOLVE THE RESULTING EQUATIONS.
Step 5 CHECK THE SOLUTIONS IN THE ORIGINAL EQUATION.
Objective 7: Solve quadratic equations by the square root property
Solving Quadratic Equations by the Square Root Property
Objective 7Solve quadratic equations by the square root property.
Watch Video
Quadratic equations of the form where u is an algebraic expression and d is a nonzero real number, can be solved by the square root property. First, isolate the squared expression on one side of the equation and the number d on the other side. Then take the square root of both sides. Remember, there are two numbers whose square is d. One number is and one is
We can use factoring to verify that has these two solutions.
Because the solutions differ only in sign, we can write them in abbreviated notation as We read this as “u equals positive or negative the square root of d” or “u equals plus or minus the square root of d.”
Now that we have verified these solutions, we can solve directly by taking square roots. This process is called the square root property.
The Square Root Property
If u is an algebraic expression and d is a nonzero number, then has exactly two solutions:
Equivalently,
Before you can apply the square root property, a squared expression must be isolated on one side of the equation.
Example 8 Solving Quadratic Equations by the Square Root Property
Solve by the square root property:
Solution
To apply the square root property, we need a squared expression by itself on one side of the equation.
By checking both proposed solutions in the original equation, we can confirm that the solution set is or
By checking both values in the original equation, we can confirm that the solution set is or
Objective 8: Solve quadratic equations by completing the square
Solving Quadratic Equations by Completing the Square
Objective 8Solve quadratic equations by completing the square.
Watch Video
How do we solve an equation in the form if the trinomial cannot be factored? We cannot use the zero-product principle in such a case. However, we can convert the equation into an equivalent equation that can be solved using the square root property. This is accomplished by completing the square.
Completing the Square
If is a binomial, then by adding which is the square of half the coefficient of x, a perfect square trinomial will result. That is,
We can solve any quadratic equation by completing the square and then applying the square root property.
Solving a Quadratic Equation by Completing the Square
To solve by completing the square:
If a, the leading coefficient, is not 1, divide both sides by a. This makes the coefficient of the -term 1.
Isolate the variable terms on one side of the equation and the constant term on the other side of the equation.
Complete the square.
Add the square of half the coefficient of x to both sides of the equation.
Factor the resulting perfect square trinomial.
Use the square root property and solve for x.
Example 9 Solving a Quadratic Equation by Completing the Square
Solve by completing the square:
Solution
Step 1 DIVIDE BOTH SIDES BY a TO MAKE THE LEADING COEFFICIENT 1.
Step 2 ISOLATE THE VARIABLE TERMS ON ONE SIDE OF THE EQUATION AND THE CONSTANT TERM ON THE OTHER SIDE.
Step 3(a)COMPLETE THE SQUARE. ADD THE SQUARE OF HALF THE COEFFICIENT OF x TO BOTH SIDES OF THE EQUATION.
Step 3(b)FACTOR THE RESULTING PERFECT SQUARE TRINOMIAL.
Step 4 USE THE SQUARE ROOT PROPERTY AND SOLVE FOR x.
Objective 9: Solve quadratic equations using the quadratic formula
Solving Quadratic Equations Using the Quadratic Formula
Objective 9Solve quadratic equations using the quadratic formula.
Watch Video
We can use the method of completing the square to derive a formula that can be used to solve all quadratic equations. The derivation given below also shows a particular quadratic equation, to specifically illustrate each of the steps.
The formula shown at the bottom of the left column is called the quadratic formula. A similar proof shows that the same formula can be used to solve quadratic equations if a, the coefficient of the is negative.
The Quadratic Formula
The solutions of a quadratic equation in standard form with are given by the quadratic formula:
To use the quadratic formula, rewrite the quadratic equation in standard form if necessary. Then determine the numerical values for a (the coefficient of the b (the coefficient of the ), and c (the constant term). Substitute the values of and c into the quadratic formula and evaluate the expression. The ± sign indicates that there are two (not necessarily distinct) solutions of the equation.
Example 10 Solving a Quadratic Equation Using the Quadratic Formula
Solve using the quadratic formula:
Solution
The given equation is in standard form. Begin by identifying the values for and
Substituting these values into the quadratic formula and simplifying gives the equation’s solutions.
Objective 9: Solve quadratic equations using the quadratic formula
Solving Quadratic Equations Using the Quadratic Formula
Objective 9Solve quadratic equations using the quadratic formula.
Watch Video
We can use the method of completing the square to derive a formula that can be used to solve all quadratic equations. The derivation given below also shows a particular quadratic equation, to specifically illustrate each of the steps.
The formula shown at the bottom of the left column is called the quadratic formula. A similar proof shows that the same formula can be used to solve quadratic equations if a, the coefficient of the is negative.
The Quadratic Formula
The solutions of a quadratic equation in standard form with are given by the quadratic formula:
To use the quadratic formula, rewrite the quadratic equation in standard form if necessary. Then determine the numerical values for a (the coefficient of the b (the coefficient of the ), and c (the constant term). Substitute the values of and c into the quadratic formula and evaluate the expression. The ± sign indicates that there are two (not necessarily distinct) solutions of the equation.
Example 10 Solving a Quadratic Equation Using the Quadratic Formula
Solve using the quadratic formula:
Solution
The given equation is in standard form. Begin by identifying the values for and
Substituting these values into the quadratic formula and simplifying gives the equation’s solutions.
Objective 10: Use the discriminant to determine the number and type of solutions of quadratic equations
Quadratic Equations and the Discriminant
Objective 10Use the discriminant to determine the number and type of solutions of quadratic equations.
Watch Video
The quantity which appears under the radical sign in the quadratic formula, is called the discriminant.Table P.4 shows how the discriminant of the quadratic equation determines the number and type of solutions.
Table P.4 The Discriminant and the Kinds of Solutions to
Discriminant
Kinds of Solutions to
Two unequal real solutions: If a, b, and c are rational numbers and the discriminant is a perfect square, the solutions are rational. If the discriminant is not a perfect square, the solutions are irrational.
One solution (a repeated solution) that is a real number: If a, b, and c are rational numbers, the repeated solution is also a rational number.
No real solutions
Example 11 Using the Discriminant
Compute the discriminant of What does the discriminant indicate about the number and type of solutions?
Solution
Begin by identifying the values for a, b, and c.
Substitute and compute the discriminant:
The discriminant is 48. Because the discriminant is positive, the equation has two unequal real solutions.
Compute the discriminant of What does the discriminant indicate about the number and type of solutions?
Objective 11: Determine the most efficient method to use when solving a quadratic equation
Determining Which Method to Use
Objective 11Determine the most efficient method to use when solving a quadratic equation.
Watch Video
All quadratic equations can be solved by the quadratic formula. However, if an equation is in the form such as or it is faster to use the square root property, taking the square root of both sides. If the equation is not in the form write the quadratic equation in standard form Try to solve the equation by factoring. If cannot be factored, then solve the quadratic equation by the quadratic formula.
Because we used the method of completing the square to derive the quadratic formula, we no longer need it for solving quadratic equations. However, we will use completing the square later in the book to help graph other kinds of equations.
Table P.5 summarizes our observations about which technique to use when solving a quadratic equation.
Table P.5 Determining the Most Efficient Technique to Use When Solving a Quadratic Equation
A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. An example of a radical equation is
We solve by squaring both sides:
The proposed solution, 81, can be checked in the original equation, Because the solution is 81 and the solution set is
In general, we solve radical equations with square roots by squaring both sides of the equation. We solve radical equations with nth roots by raising both sides of the equation to the nth power. Unfortunately, if n is even, all the solutions of the equation raised to the even power may not be solutions of the original equation. Consider, for example, the equation
If we square both sides, we obtain
Solving this equation using the square root property, we obtain
The new equation has two solutions, and 4. By contrast, only 4 is a solution of the original equation, For this reason, when raising both sides of an equation to an even power, always check proposed solutions in the original equation.
Here is a general method for solving radical equations with nth roots:
Solving Radical Equations Containing nth Roots
If necessary, arrange terms so that one radical is isolated on one side of the equation.
Raise both sides of the equation to the nth power to eliminate the isolated nth root.
Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2.
Check all proposed solutions in the original equation.
Extra solutions may be introduced when you raise both sides of a radical equation to an even power. Such solutions, which are not solutions of the given equation, are called extraneous solutions or extraneous roots.
Example 12 Solving a Radical Equation
Solve:
Solution
Step 1 ISOLATE A RADICAL ON ONE SIDE. We isolate the radical, by subtracting 2 from both sides.
Step 2 RAISE BOTH SIDES TO THE nTH POWER. Because n, the index of the radical in the equation is 2, we square both sides.
Step 3 SOLVE THE RESULTING EQUATION. Because of the the resulting equation is a quadratic equation. We can obtain 0 on the left side by subtracting 2x and adding 1 on both sides.
Step 4 CHECK THE PROPOSED SOLUTIONS IN THE ORIGINAL EQUATION.
Check 1:
Check 5:
Thus, 1 is an extraneous solution. The only solution is 5, and the solution set is
A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. An example of a radical equation is
We solve by squaring both sides:
The proposed solution, 81, can be checked in the original equation, Because the solution is 81 and the solution set is
In general, we solve radical equations with square roots by squaring both sides of the equation. We solve radical equations with nth roots by raising both sides of the equation to the nth power. Unfortunately, if n is even, all the solutions of the equation raised to the even power may not be solutions of the original equation. Consider, for example, the equation
If we square both sides, we obtain
Solving this equation using the square root property, we obtain
The new equation has two solutions, and 4. By contrast, only 4 is a solution of the original equation, For this reason, when raising both sides of an equation to an even power, always check proposed solutions in the original equation.
Here is a general method for solving radical equations with nth roots:
Solving Radical Equations Containing nth Roots
If necessary, arrange terms so that one radical is isolated on one side of the equation.
Raise both sides of the equation to the nth power to eliminate the isolated nth root.
Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2.
Check all proposed solutions in the original equation.
Extra solutions may be introduced when you raise both sides of a radical equation to an even power. Such solutions, which are not solutions of the given equation, are called extraneous solutions or extraneous roots.
Example 12 Solving a Radical Equation
Solve:
Solution
Step 1 ISOLATE A RADICAL ON ONE SIDE. We isolate the radical, by subtracting 2 from both sides.
Step 2 RAISE BOTH SIDES TO THE nTH POWER. Because n, the index of the radical in the equation is 2, we square both sides.
Step 3 SOLVE THE RESULTING EQUATION. Because of the the resulting equation is a quadratic equation. We can obtain 0 on the left side by subtracting 2x and adding 1 on both sides.
Step 4 CHECK THE PROPOSED SOLUTIONS IN THE ORIGINAL EQUATION.
Check 1:
Check 5:
Thus, 1 is an extraneous solution. The only solution is 5, and the solution set is
Squaring the left side and simplifying results in ______. Squaring the right side and simplifying results in ______.
P.7: Exercise Set
P.7 Exercise Set
Practice Exercises
In Exercises 1–16, solve and check each linear equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Exercises 17–26 contain rational equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–42, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe?
27. for P
28. for r
29. for p
30. for M
31. for a
32. for b
33. for r
34. for t
35. for S
36. for r
37. for I
38. for h
39. for f
40. for
41. for
42. for
In Exercises 43–54, solve each absolute value equation or indicate the equation has no solution.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
In Exercises 55–60, solve each quadratic equation by factoring.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, solve each quadratic equation by the square root property.
61.
62.
63.
64.
65.
66.
In Exercises 67–74, solve each quadratic equation by completing the square.
67.
68.
69.
70.
71.
72.
73.
74.
In Exercises 75–82, solve each quadratic equation using the quadratic formula.
75.
76.
77.
78.
79.
80.
81.
82.
Compute the discriminant of each equation in Exercises 83–90. What does the discriminant indicate about the number and type of solutions?
83.
84.
85.
86.
87.
88.
89.
90.
In Exercises 91–114, solve each quadratic equation by the method of your choice.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
In Exercises 115–124, solve each radical equation. Check all proposed solutions.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
Practice PLUS
In Exercises 125–134, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
In Exercises 135–136, list all numbers that must be excluded from the domain of each rational expression.
135.
136.
Application Exercises
Grade Inflation. The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school.
The data displayed by the bar graph can be described by the mathematical model
where x is the number of years after 1980 and p is the percentage of U.S. college freshmen who had an average grade of A in high school. Use this information to solve Exercises 137–138.
137.
According to the formula, in 2010, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 57% of U.S. college freshmen will have had an average grade of A in high school.
138.
According to the formula, in 2000, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 65% of U.S. college freshmen will have had an average grade of A in high school.
139. A company wants to increase the 10% peroxide content of its product by adding pure peroxide (100% peroxide). If x liters of pure peroxide are added to 500 liters of its 10% solution, the concentration, C, of the new mixture is given by
How many liters of pure peroxide should be added to produce a new product that is 28% peroxide?
140. Suppose that x liters of pure acid are added to 200 liters of a 35% acid solution.
Write a formula that gives the concentration, C, of the new mixture. (Hint: See Exercise 139.)
How many liters of pure acid should be added to produce a new mixture that is 74% acid?
A driver’s age has something to do with his or her chance of getting into a fatal car crash. The bar graph shows the number of fatal vehicle crashes per 100 million miles driven for drivers of various age groups. For example, 25-year-old drivers are involved in 4.1 fatal crashes per 100 million miles driven. Thus, when a group of 25-year-old Americans have driven a total of 100 million miles, approximately 4 have been in accidents in which someone died.
The number of fatal vehicle crashes per 100 million miles, N, for drivers of age x can be modeled by the formula
Use the formula to solve Exercises 141–142.
141. What age groups are expected to be involved in 3 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
142. What age groups are expected to be involved in 10 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
The graphs show the percentage of jobs in the U.S. labor force held by men and by women from 1970 through 2015. Exercises 143–144 are based on the data displayed by the graphs.
models the percentage of jobs in the U.S. labor force, p, held by women t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by women in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by women in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 51% of jobs in the U.S. labor force be held by women? Round to the nearest year.
144. The formula
models the percentage of jobs in the U.S. labor force, p, held by men t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by men in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by men in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 49% of jobs in the U.S. labor force be held by men? Round to the nearest year.
Explaining the Concepts
145. What is a linear equation in one variable? Give an example of this type of equation.
146. Explain how to determine the restrictions on the variable for the equation
147. What does it mean to solve a formula for a variable?
148. Explain how to solve an equation involving absolute value.
149. Why does the procedure that you explained in Exercise 148 not apply to the equation What is the solution set for this equation?
150. What is a quadratic equation?
151. Explain how to solve using factoring and the zero-product principle.
152. Explain how to solve by completing the square.
153. Explain how to solve using the quadratic formula.
154. How is the quadratic formula derived?
155. What is the discriminant and what information does it provide about a quadratic equation?
156. If you are given a quadratic equation, how do you determine which method to use to solve it?
157. In solving why is it a good idea to isolate the radical term? What if we don’t do this and simply square each side? Describe what happens.
158. What is an extraneous solution to a radical equation?
Critical Thinking Exercises
Make Sense? In Exercises 159–162, determine whether each statement makes sense or does not make sense, and explain your reasoning.
159. The model describes the percentage of college freshmen with an A average in high school, p, x years after 1980, so I have to solve a linear equation to determine the percentage of college freshmen with an A average in high school in 2020.
160. Although I can solve by first subtracting from both sides, I find it easier to begin by multiplying both sides by 20, the least common denominator.
161. Because I want to solve fairly quickly, I’ll use the quadratic formula.
162. When checking a radical equation’s proposed solution, I can substitute into the original equation or any equation that is part of the solution process.
In Exercises 163–166, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
163. The equation is equivalent to
164. Every quadratic equation has two distinct numbers in its solution set.
165. The equations and are equivalent.
166. The equation cannot be solved by the quadratic formula.
167. Find b such that will have a solution set given by
168. Write a quadratic equation in standard form whose solution set is
169. Solve for
170. Solve for
Preview Exercises
Exercises 171–173 will help you prepare for the material covered in the next section.
171. Jane’s salary exceeds Jim’s by $150 per week. If x represents Jim’s weekly salary, write an algebraic expression that models Jane’s weekly salary.
172. A convenience store sells a refillable stainless-steel travel mug for $20. Customers who buy the mug can refill it with coffee for just $0.99 on each visit. Write an algebraic expression that models the total cost of the mug and x refills.
173. If the width of a rectangle is represented by x and the length is represented by write a simplified algebraic expression that models the rectangle’s perimeter.
P.7: Exercise Set
P.7 Exercise Set
Practice Exercises
In Exercises 1–16, solve and check each linear equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Exercises 17–26 contain rational equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–42, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe?
27. for P
28. for r
29. for p
30. for M
31. for a
32. for b
33. for r
34. for t
35. for S
36. for r
37. for I
38. for h
39. for f
40. for
41. for
42. for
In Exercises 43–54, solve each absolute value equation or indicate the equation has no solution.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
In Exercises 55–60, solve each quadratic equation by factoring.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, solve each quadratic equation by the square root property.
61.
62.
63.
64.
65.
66.
In Exercises 67–74, solve each quadratic equation by completing the square.
67.
68.
69.
70.
71.
72.
73.
74.
In Exercises 75–82, solve each quadratic equation using the quadratic formula.
75.
76.
77.
78.
79.
80.
81.
82.
Compute the discriminant of each equation in Exercises 83–90. What does the discriminant indicate about the number and type of solutions?
83.
84.
85.
86.
87.
88.
89.
90.
In Exercises 91–114, solve each quadratic equation by the method of your choice.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
In Exercises 115–124, solve each radical equation. Check all proposed solutions.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
Practice PLUS
In Exercises 125–134, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
In Exercises 135–136, list all numbers that must be excluded from the domain of each rational expression.
135.
136.
Application Exercises
Grade Inflation. The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school.
The data displayed by the bar graph can be described by the mathematical model
where x is the number of years after 1980 and p is the percentage of U.S. college freshmen who had an average grade of A in high school. Use this information to solve Exercises 137–138.
137.
According to the formula, in 2010, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 57% of U.S. college freshmen will have had an average grade of A in high school.
138.
According to the formula, in 2000, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 65% of U.S. college freshmen will have had an average grade of A in high school.
139. A company wants to increase the 10% peroxide content of its product by adding pure peroxide (100% peroxide). If x liters of pure peroxide are added to 500 liters of its 10% solution, the concentration, C, of the new mixture is given by
How many liters of pure peroxide should be added to produce a new product that is 28% peroxide?
140. Suppose that x liters of pure acid are added to 200 liters of a 35% acid solution.
Write a formula that gives the concentration, C, of the new mixture. (Hint: See Exercise 139.)
How many liters of pure acid should be added to produce a new mixture that is 74% acid?
A driver’s age has something to do with his or her chance of getting into a fatal car crash. The bar graph shows the number of fatal vehicle crashes per 100 million miles driven for drivers of various age groups. For example, 25-year-old drivers are involved in 4.1 fatal crashes per 100 million miles driven. Thus, when a group of 25-year-old Americans have driven a total of 100 million miles, approximately 4 have been in accidents in which someone died.
The number of fatal vehicle crashes per 100 million miles, N, for drivers of age x can be modeled by the formula
Use the formula to solve Exercises 141–142.
141. What age groups are expected to be involved in 3 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
142. What age groups are expected to be involved in 10 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
The graphs show the percentage of jobs in the U.S. labor force held by men and by women from 1970 through 2015. Exercises 143–144 are based on the data displayed by the graphs.
models the percentage of jobs in the U.S. labor force, p, held by women t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by women in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by women in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 51% of jobs in the U.S. labor force be held by women? Round to the nearest year.
144. The formula
models the percentage of jobs in the U.S. labor force, p, held by men t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by men in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by men in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 49% of jobs in the U.S. labor force be held by men? Round to the nearest year.
Explaining the Concepts
145. What is a linear equation in one variable? Give an example of this type of equation.
146. Explain how to determine the restrictions on the variable for the equation
147. What does it mean to solve a formula for a variable?
148. Explain how to solve an equation involving absolute value.
149. Why does the procedure that you explained in Exercise 148 not apply to the equation What is the solution set for this equation?
150. What is a quadratic equation?
151. Explain how to solve using factoring and the zero-product principle.
152. Explain how to solve by completing the square.
153. Explain how to solve using the quadratic formula.
154. How is the quadratic formula derived?
155. What is the discriminant and what information does it provide about a quadratic equation?
156. If you are given a quadratic equation, how do you determine which method to use to solve it?
157. In solving why is it a good idea to isolate the radical term? What if we don’t do this and simply square each side? Describe what happens.
158. What is an extraneous solution to a radical equation?
Critical Thinking Exercises
Make Sense? In Exercises 159–162, determine whether each statement makes sense or does not make sense, and explain your reasoning.
159. The model describes the percentage of college freshmen with an A average in high school, p, x years after 1980, so I have to solve a linear equation to determine the percentage of college freshmen with an A average in high school in 2020.
160. Although I can solve by first subtracting from both sides, I find it easier to begin by multiplying both sides by 20, the least common denominator.
161. Because I want to solve fairly quickly, I’ll use the quadratic formula.
162. When checking a radical equation’s proposed solution, I can substitute into the original equation or any equation that is part of the solution process.
In Exercises 163–166, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
163. The equation is equivalent to
164. Every quadratic equation has two distinct numbers in its solution set.
165. The equations and are equivalent.
166. The equation cannot be solved by the quadratic formula.
167. Find b such that will have a solution set given by
168. Write a quadratic equation in standard form whose solution set is
169. Solve for
170. Solve for
Preview Exercises
Exercises 171–173 will help you prepare for the material covered in the next section.
171. Jane’s salary exceeds Jim’s by $150 per week. If x represents Jim’s weekly salary, write an algebraic expression that models Jane’s weekly salary.
172. A convenience store sells a refillable stainless-steel travel mug for $20. Customers who buy the mug can refill it with coffee for just $0.99 on each visit. Write an algebraic expression that models the total cost of the mug and x refills.
173. If the width of a rectangle is represented by x and the length is represented by write a simplified algebraic expression that models the rectangle’s perimeter.
P.7: Exercise Set
P.7 Exercise Set
Practice Exercises
In Exercises 1–16, solve and check each linear equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Exercises 17–26 contain rational equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–42, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe?
27. for P
28. for r
29. for p
30. for M
31. for a
32. for b
33. for r
34. for t
35. for S
36. for r
37. for I
38. for h
39. for f
40. for
41. for
42. for
In Exercises 43–54, solve each absolute value equation or indicate the equation has no solution.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
In Exercises 55–60, solve each quadratic equation by factoring.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, solve each quadratic equation by the square root property.
61.
62.
63.
64.
65.
66.
In Exercises 67–74, solve each quadratic equation by completing the square.
67.
68.
69.
70.
71.
72.
73.
74.
In Exercises 75–82, solve each quadratic equation using the quadratic formula.
75.
76.
77.
78.
79.
80.
81.
82.
Compute the discriminant of each equation in Exercises 83–90. What does the discriminant indicate about the number and type of solutions?
83.
84.
85.
86.
87.
88.
89.
90.
In Exercises 91–114, solve each quadratic equation by the method of your choice.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
In Exercises 115–124, solve each radical equation. Check all proposed solutions.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
Practice PLUS
In Exercises 125–134, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
In Exercises 135–136, list all numbers that must be excluded from the domain of each rational expression.
135.
136.
Application Exercises
Grade Inflation. The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school.
The data displayed by the bar graph can be described by the mathematical model
where x is the number of years after 1980 and p is the percentage of U.S. college freshmen who had an average grade of A in high school. Use this information to solve Exercises 137–138.
137.
According to the formula, in 2010, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 57% of U.S. college freshmen will have had an average grade of A in high school.
138.
According to the formula, in 2000, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 65% of U.S. college freshmen will have had an average grade of A in high school.
139. A company wants to increase the 10% peroxide content of its product by adding pure peroxide (100% peroxide). If x liters of pure peroxide are added to 500 liters of its 10% solution, the concentration, C, of the new mixture is given by
How many liters of pure peroxide should be added to produce a new product that is 28% peroxide?
140. Suppose that x liters of pure acid are added to 200 liters of a 35% acid solution.
Write a formula that gives the concentration, C, of the new mixture. (Hint: See Exercise 139.)
How many liters of pure acid should be added to produce a new mixture that is 74% acid?
A driver’s age has something to do with his or her chance of getting into a fatal car crash. The bar graph shows the number of fatal vehicle crashes per 100 million miles driven for drivers of various age groups. For example, 25-year-old drivers are involved in 4.1 fatal crashes per 100 million miles driven. Thus, when a group of 25-year-old Americans have driven a total of 100 million miles, approximately 4 have been in accidents in which someone died.
The number of fatal vehicle crashes per 100 million miles, N, for drivers of age x can be modeled by the formula
Use the formula to solve Exercises 141–142.
141. What age groups are expected to be involved in 3 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
142. What age groups are expected to be involved in 10 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
The graphs show the percentage of jobs in the U.S. labor force held by men and by women from 1970 through 2015. Exercises 143–144 are based on the data displayed by the graphs.
models the percentage of jobs in the U.S. labor force, p, held by women t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by women in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by women in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 51% of jobs in the U.S. labor force be held by women? Round to the nearest year.
144. The formula
models the percentage of jobs in the U.S. labor force, p, held by men t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by men in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by men in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 49% of jobs in the U.S. labor force be held by men? Round to the nearest year.
Explaining the Concepts
145. What is a linear equation in one variable? Give an example of this type of equation.
146. Explain how to determine the restrictions on the variable for the equation
147. What does it mean to solve a formula for a variable?
148. Explain how to solve an equation involving absolute value.
149. Why does the procedure that you explained in Exercise 148 not apply to the equation What is the solution set for this equation?
150. What is a quadratic equation?
151. Explain how to solve using factoring and the zero-product principle.
152. Explain how to solve by completing the square.
153. Explain how to solve using the quadratic formula.
154. How is the quadratic formula derived?
155. What is the discriminant and what information does it provide about a quadratic equation?
156. If you are given a quadratic equation, how do you determine which method to use to solve it?
157. In solving why is it a good idea to isolate the radical term? What if we don’t do this and simply square each side? Describe what happens.
158. What is an extraneous solution to a radical equation?
Critical Thinking Exercises
Make Sense? In Exercises 159–162, determine whether each statement makes sense or does not make sense, and explain your reasoning.
159. The model describes the percentage of college freshmen with an A average in high school, p, x years after 1980, so I have to solve a linear equation to determine the percentage of college freshmen with an A average in high school in 2020.
160. Although I can solve by first subtracting from both sides, I find it easier to begin by multiplying both sides by 20, the least common denominator.
161. Because I want to solve fairly quickly, I’ll use the quadratic formula.
162. When checking a radical equation’s proposed solution, I can substitute into the original equation or any equation that is part of the solution process.
In Exercises 163–166, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
163. The equation is equivalent to
164. Every quadratic equation has two distinct numbers in its solution set.
165. The equations and are equivalent.
166. The equation cannot be solved by the quadratic formula.
167. Find b such that will have a solution set given by
168. Write a quadratic equation in standard form whose solution set is
169. Solve for
170. Solve for
Preview Exercises
Exercises 171–173 will help you prepare for the material covered in the next section.
171. Jane’s salary exceeds Jim’s by $150 per week. If x represents Jim’s weekly salary, write an algebraic expression that models Jane’s weekly salary.
172. A convenience store sells a refillable stainless-steel travel mug for $20. Customers who buy the mug can refill it with coffee for just $0.99 on each visit. Write an algebraic expression that models the total cost of the mug and x refills.
173. If the width of a rectangle is represented by x and the length is represented by write a simplified algebraic expression that models the rectangle’s perimeter.
P.7: Exercise Set
P.7 Exercise Set
Practice Exercises
In Exercises 1–16, solve and check each linear equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Exercises 17–26 contain rational equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–42, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe?
27. for P
28. for r
29. for p
30. for M
31. for a
32. for b
33. for r
34. for t
35. for S
36. for r
37. for I
38. for h
39. for f
40. for
41. for
42. for
In Exercises 43–54, solve each absolute value equation or indicate the equation has no solution.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
In Exercises 55–60, solve each quadratic equation by factoring.
55.
56.
57.
58.
59.
60.
In Exercises 61–66, solve each quadratic equation by the square root property.
61.
62.
63.
64.
65.
66.
In Exercises 67–74, solve each quadratic equation by completing the square.
67.
68.
69.
70.
71.
72.
73.
74.
In Exercises 75–82, solve each quadratic equation using the quadratic formula.
75.
76.
77.
78.
79.
80.
81.
82.
Compute the discriminant of each equation in Exercises 83–90. What does the discriminant indicate about the number and type of solutions?
83.
84.
85.
86.
87.
88.
89.
90.
In Exercises 91–114, solve each quadratic equation by the method of your choice.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
In Exercises 115–124, solve each radical equation. Check all proposed solutions.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
Practice PLUS
In Exercises 125–134, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
In Exercises 135–136, list all numbers that must be excluded from the domain of each rational expression.
135.
136.
Application Exercises
Grade Inflation. The bar graph shows the percentage of U.S. college freshmen with an average grade of A in high school.
The data displayed by the bar graph can be described by the mathematical model
where x is the number of years after 1980 and p is the percentage of U.S. college freshmen who had an average grade of A in high school. Use this information to solve Exercises 137–138.
137.
According to the formula, in 2010, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 57% of U.S. college freshmen will have had an average grade of A in high school.
138.
According to the formula, in 2000, what percentage of U.S. college freshmen had an average grade of A in high school? Does this underestimate or overestimate the percent displayed by the bar graph? By how much?
If trends shown by the formula continue, project when 65% of U.S. college freshmen will have had an average grade of A in high school.
139. A company wants to increase the 10% peroxide content of its product by adding pure peroxide (100% peroxide). If x liters of pure peroxide are added to 500 liters of its 10% solution, the concentration, C, of the new mixture is given by
How many liters of pure peroxide should be added to produce a new product that is 28% peroxide?
140. Suppose that x liters of pure acid are added to 200 liters of a 35% acid solution.
Write a formula that gives the concentration, C, of the new mixture. (Hint: See Exercise 139.)
How many liters of pure acid should be added to produce a new mixture that is 74% acid?
A driver’s age has something to do with his or her chance of getting into a fatal car crash. The bar graph shows the number of fatal vehicle crashes per 100 million miles driven for drivers of various age groups. For example, 25-year-old drivers are involved in 4.1 fatal crashes per 100 million miles driven. Thus, when a group of 25-year-old Americans have driven a total of 100 million miles, approximately 4 have been in accidents in which someone died.
The number of fatal vehicle crashes per 100 million miles, N, for drivers of age x can be modeled by the formula
Use the formula to solve Exercises 141–142.
141. What age groups are expected to be involved in 3 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
142. What age groups are expected to be involved in 10 fatal crashes per 100 million miles driven? How well does the formula model the trend in the actual data shown in the bar graph?
The graphs show the percentage of jobs in the U.S. labor force held by men and by women from 1970 through 2015. Exercises 143–144 are based on the data displayed by the graphs.
models the percentage of jobs in the U.S. labor force, p, held by women t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by women in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by women in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 51% of jobs in the U.S. labor force be held by women? Round to the nearest year.
144. The formula
models the percentage of jobs in the U.S. labor force, p, held by men t years after 1970.
Use the appropriate graph to estimate the percentage of jobs in the U.S. labor force held by men in 2010. Give your estimation to the nearest percent.
Use the mathematical model to determine the percentage of jobs in the U.S. labor force held by men in 2010. Round to the nearest tenth of a percent.
According to the formula, when will 49% of jobs in the U.S. labor force be held by men? Round to the nearest year.
Explaining the Concepts
145. What is a linear equation in one variable? Give an example of this type of equation.
146. Explain how to determine the restrictions on the variable for the equation
147. What does it mean to solve a formula for a variable?
148. Explain how to solve an equation involving absolute value.
149. Why does the procedure that you explained in Exercise 148 not apply to the equation What is the solution set for this equation?
150. What is a quadratic equation?
151. Explain how to solve using factoring and the zero-product principle.
152. Explain how to solve by completing the square.
153. Explain how to solve using the quadratic formula.
154. How is the quadratic formula derived?
155. What is the discriminant and what information does it provide about a quadratic equation?
156. If you are given a quadratic equation, how do you determine which method to use to solve it?
157. In solving why is it a good idea to isolate the radical term? What if we don’t do this and simply square each side? Describe what happens.
158. What is an extraneous solution to a radical equation?
Critical Thinking Exercises
Make Sense? In Exercises 159–162, determine whether each statement makes sense or does not make sense, and explain your reasoning.
159. The model describes the percentage of college freshmen with an A average in high school, p, x years after 1980, so I have to solve a linear equation to determine the percentage of college freshmen with an A average in high school in 2020.
160. Although I can solve by first subtracting from both sides, I find it easier to begin by multiplying both sides by 20, the least common denominator.
161. Because I want to solve fairly quickly, I’ll use the quadratic formula.
162. When checking a radical equation’s proposed solution, I can substitute into the original equation or any equation that is part of the solution process.
In Exercises 163–166, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
163. The equation is equivalent to
164. Every quadratic equation has two distinct numbers in its solution set.
165. The equations and are equivalent.
166. The equation cannot be solved by the quadratic formula.
167. Find b such that will have a solution set given by
168. Write a quadratic equation in standard form whose solution set is
169. Solve for
170. Solve for
Preview Exercises
Exercises 171–173 will help you prepare for the material covered in the next section.
171. Jane’s salary exceeds Jim’s by $150 per week. If x represents Jim’s weekly salary, write an algebraic expression that models Jane’s weekly salary.
172. A convenience store sells a refillable stainless-steel travel mug for $20. Customers who buy the mug can refill it with coffee for just $0.99 on each visit. Write an algebraic expression that models the total cost of the mug and x refills.
173. If the width of a rectangle is represented by x and the length is represented by write a simplified algebraic expression that models the rectangle’s perimeter.
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
Objective 1: Use equations to solve problems
Problem Solving with Equations
Objective 1Use equations to solve problems.
Watch Video
We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. This means that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we must understand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will use in solving word problems:
Strategy for Solving Word Problems
Step 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable) represent one of the unknown quantities in the problem.
Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.
Step 3 Write an equation in x that models the verbal conditions of the problem.
Step 4 Solve the equation and answer the problem’s question.
Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.
Example 1 Education Pays Off
The graph in Figure P.14 shows median yearly earnings in the United States by highest educational attainment. It certainly looks like you can make more money with more education. However, workers who make more money pay a higher percentage of income in taxes, so it is wise to consider after-tax earnings when deciding if that higher degree will pay off.
The median yearly after-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $11 thousand. The median yearly after-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $21 thousand. Combined, three full-time workers with each of these degrees earn $149 thousand after taxes. Find the median yearly after-tax income of full-time workers with each of these levels of education.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about after-tax incomes of full-time workers with bachelor’s degrees and master’s degrees: They exceed the after-tax income of a full-time worker with an associate’s degree by $11 thousand and $21 thousand, respectively. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because a full-time worker with a bachelor’s degree earns $11 thousand more after taxes than a full-time worker with an associate’s degree, let
Because a full-time worker with a master’s degree earns $21 thousand more after taxes than a full-time worker with an associate’s degree, let
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Combined, three full-time workers with each of these degrees earn $149 thousand.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because we isolated the variable in the model and obtained ,
Full-time workers with associate’s degrees earn $39 thousand per year after taxes, full-time workers with bachelor’s degrees earn $50 thousand per year after taxes, and full-time workers with master’s degrees earn $60 thousand per year after taxes.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that combined, three full-time workers with each of these educational attainments earn $149 thousand after taxes. Using the incomes we determined in step 4, the sum is
The median yearly before-tax income of a full-time worker with a bachelor’s degree exceeds that of a full-time worker with an associate’s degree by $15 thousand. The median yearly before-tax income of a full-time worker with a master’s degree exceeds that of a full-time worker with an associate’s degree by $30 thousand. Combined, three full-time workers with each of these educational attainments earn $195 thousand before taxes. Find the median yearly before-tax income of full-time workers with each of these levels of education. (These incomes are illustrated by the bar graph on the previous page.)
Example 2 Modeling Attitudes of College Freshmen
Your author teaching math in 1969
Researchers have surveyed college freshmen every year since 1969. Figure P.15 shows that attitudes about some life goals have changed dramatically over the years. In particular, the freshman class of 2018 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college students considered “being well-off financially” essential or very important. For the period from 1969 through 2018, this percentage increased by approximately 0.8 each year. If this trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this life objective essential or very important. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Using current trends, by approximately 73 years after 1969, or in 2042, all freshmen will consider “being well-off financially” essential or very important.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that all freshmen (100%, represented by 100 using the model) will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42, by 0.8 each year for 73 years, our proposed solution?
This verifies that using trends shown in Figure P.15, all first-year college students will consider the objective of being well-off financially essential or very important approximately 73 years after 1969, or in 2042.
Figure P.15 on the previous page shows that the freshman class of 2018 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of the freshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.8 each year. If this trend continues, by which year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?
Example 3 Modeling Options for a Toll
It’s time to decide how you’re going to pay the bridge toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 each time you cross the bridge plus a $3 administrative fee for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. Let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The cost with the transponder is the cost of the transponder, $20, plus the toll, $3.25, times the number of crossings, x. For toll-by-plate, you must pay both the toll, $4.25, and the administrative fee, $3, each time you cross the bridge. The cost for toll-by-plate is the sum of the toll and the administrative fee, or $7.25, times the number of crossings, x. We want to know when these two costs are equal.
Step 4. SOLVE THE EQUATION AND ANSWER THE QUESTION.
Because x represents the number of times you cross the bridge, the total cost of toll-by-plate is the same as the total cost with the transponder for 5 crossings.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the cost of toll-by-plate should be the same as the cost with the transponder. Let’s see if they are the same with 5 bridge crossings per month.
You drive up to a toll plaza and find booths with attendants where you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge for the costs of the two options to be the same.
Example 4 A Price Reduction on Wireless Headphones
Your favorite electronics retailer is having a terrific sale on noise cancelling wireless headphones. (You’ll finally be able to listen to music in peace!) After a 40% price reduction, you purchase headphones for $192. What was the price of the headphones before the reduction?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. There are no other unknown quantities to find, so we can skip this step.
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. The headphones’ original price minus the 40% reduction is the reduced price, $192.
The headphones’ price before the reduction was $320.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The price before the reduction, $320, minus the 40% reduction should equal the reduced price given in the original wording, $192:
This verifies that the headphones’ price before the reduction was $320.
After a 30% price reduction, you purchase a new laptop for $840. What was the laptop’s price before the reduction?
Our next example is about simple interest. Simple interest involves interest calculated only on the amount of money that we invest, called the principal. The formula is used to find the simple interest, I, earned for one year when the principal, P, is invested at an annual interest rate, r. Dual investment problems involve different amounts of money in two or more investments, each paying a different rate.
Example 5 Solving a Dual Investment Problem
Your grandmother needs your help. She has $150,000 to invest. Part of this money is to be invested in a money market account paying 1.5% annual interest. The rest of this money is to be invested in a certificate of deposit paying 1.3% annual interest. She told you that she requires $2000 per year in extra income from the combination of these investments. How much money should be placed in each investment?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. The other quantity that we seek is the amount invested at 1.3% in the certificate of deposit. Because the total amount Grandma has to invest is $150,000 and we already used up x,
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because Grandma requires $2000 in total interest, the interest for the two investments combined must be $2000. Interest is Pr or rP for each investment.
Grandma should invest $25,000 at 1.5% and $125,000 at 1.3%.
Step 5. CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. The problem states that the total interest from the dual investments should be $2000. Can Grandma count on $2000 interest? The interest earned on $25,000 at 1.5% is ($25,000) (0.015), or $375. The interest earned on $125,000 at 1.3% is ($125,000)(0.013), or $1625. The total interest is , or $2000, exactly as it should be. You’ve made your grandmother happy. (Now if you would just visit her more often …)
You inherited $50,000 with the stipulation that for the first year the money had to be invested in two accounts paying 0.9% and 1.1% annual interest. How much did you invest at each rate if the total interest earned for the year was $515?
Solving geometry problems usually requires a knowledge of basic geometric ideas and formulas. Formulas for area, perimeter, and volume are given in Table P.6.
Table P.6 Common Formulas for Area, Perimeter, and Volume
We will be using the formula for the perimeter of a rectangle, , in our next example. The formula states that a rectangle’s perimeter is the sum of twice its length and twice its width.
Example 6 Finding the Dimensions of an American Football Field
The length of an American football field is 200 feet more than the width. If the perimeter of the field is 1040 feet, what are its dimensions?
Solution
Step 1. LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We know something about the length; the length is 200 feet more than the width. We will let
Step 2. REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the length is 200 feet more than the width, we add 200 to the width to represent the length. Thus,
Figure P.16 illustrates an American football field and its dimensions.
Figure P.16 An American football field
Step 3. WRITE AN EQUATION IN x THAT MODELS THE CONDITIONS. Because the perimeter of the field is 1040 feet,
The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?
We will use the formula for the area of a rectangle, , in our next example. The formula states that a rectangle’s area is the product of its length and its width.
Example 7 Solving a Problem Involving Landscape Design
A rectangular garden measures 80 feet by 60 feet. A large path of uniform width is to be added along both shorter sides and one longer side of the garden. The landscape designer doing the work wants to double the garden’s area with the addition of this path. How wide should the path be?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
The situation is illustrated in Figure P.17. The figure shows the original 80-by-60 foot rectangular garden and the path of width x added along both shorter sides and one longer side.
Figure P.17 The garden’s area is to be doubled by adding the path.
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because the path is added along both shorter sides and one longer side, Figure P.17 shows that
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. The area of the rectangle must be doubled by the addition of the path.
Step 4 SOLVE THE EQUATION AND ANSWER THE QUESTION.
The path cannot have a negative width. Because is geometrically impossible, we use The width of the path should be 20 feet.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. Has the landscape architect doubled the garden’s area with the 20-foot-wide path? The area of the garden is 80 feet times 60 feet, or 4800 square feet. Because and represent the length and width of the expanded rectangle,
The area of the expanded rectangle is 120 feet times 80 feet, or 9600 square feet. This is double the area of the garden, 4800 square feet, as specified by the problem’s conditions.
A rectangular garden measures 16 feet by 12 feet. A path of uniform width is to be added so as to surround the entire garden. The landscape artist doing the work wants the garden and path to cover an area of 320 square feet. How wide should the path be?
In our next example, we will be using the Pythagorean Theorem to obtain a mathematical model. The ancient Greek philosopher and mathematician Pythagoras (approximately 582–500 b.c.) founded a school whose motto was “All is number.” Pythagoras is best remembered for his work with the right triangle, a triangle with one angle measuring 90°. The side opposite the 90° angle is called the hypotenuse. The other sides are called legs. Pythagoras found that if he constructed squares on each of the legs, as well as a larger square on the hypotenuse, the sum of the areas of the smaller squares is equal to the area of the larger square. This is illustrated in Figure P.18.
Figure P.18The area of the large square equals the sum of the areas of the smaller squares.
Did you know that the size of a television screen refers to the length of its diagonal? If the length of the HDTV screen in Figure P.19 is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch?
Figure P.19
Solution
Figure P.19 shows that the length, width, and diagonal of the screen form a right triangle. The diagonal is the hypotenuse of the triangle. We use the Pythagorean Theorem with , , and solve for the screen size, c.
Because c represents the size of the screen, this dimension must be positive. We reject 32. Thus, the screen size of the HDTV is 32 inches.
Figure P.20 shows the dimensions of an old TV screen. What is the size of the screen?
Figure P.20
Example 9 Dividing the Cost of a Yacht
A group of friends agrees to share the cost of a $50,000 yacht equally. Before the purchase is made, one more person joins the group. As a result, each person’s share is reduced by $2500. How many people were in the original group?
Solution
Step 1 LET REPRESENT ONE OF THE UNKNOWN QUANTITIES. We will let
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. Because one more person joined the original group, let
Step 3 WRITE AN EQUATION IN THAT MODELS THE CONDITIONS. As a result of one more person joining the original group, each person’s share is reduced by $2500.
A group of people share equally in a $5,000,000 lottery. Before the money is divided, three more winning ticket holders are declared. As a result, each person’s share is reduced by $375,000. How many people were in the original group of winners?
P.8: Exercise Set
P.8 Exercise Set
Practice and Application Exercises
How will you spend your average life expectancy of 78 years? The bar graph shows the average number of years you will devote to each of your most time-consuming activities. Exercises 1–2 are based on the data displayed by the graph.
1. According to the U.S. Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 19. Over your lifetime, how many years will you spend on each of these activities?
2. According to the U.S. Bureau of Labor Statistics, you will devote 32 years to sleeping and eating. The number of years sleeping will exceed the number of years eating by 24. Over your lifetime, how many years will you spend on each of these activities?
The bar graph shows median yearly earnings of full-time workers in the United States for people 25 years and over for three occupations and two levels of education. Exercises 3–4 are based on the data displayed by the graph.
3. The median yearly salary of a general manager with a bachelor’s degree or higher is $31,000 less than twice that of a general manager with just a high school diploma. Combined, two managers with each of these educational attainments earn $149,300. Find the median yearly salary of general managers with each of these levels of education.
4. The median yearly salary of a retail salesperson with a bachelor’s degree or higher is $14,300 less than twice that of a retail salesperson with just a high school diploma. Combined, two salespeople with each of these educational attainments earn $79,900. Find the median yearly salary of salespeople with each of these levels of education.
Despite booming new car sales with their cha-ching sounds, the average age of vehicles on U.S. roads is not going down. The bar graph shows the average price of new cars in the United States and the average age of cars on U.S. roads for two selected years. Exercises 5–6 are based on the information displayed by the graph.
5. In 2019, the average price of a new car was $38,900. For the period shown, new-car prices increased by approximately $800 per year. If this trend continues, how many years after 2019 will the price of a new car average $44,500? In which year will this occur?
6. In 2019, the average age of cars on U.S. roads was 11.8 years. For the period shown, this average age increased by approximately 0.15 year per year. If this trend continues, how many years after 2019 will the average age of vehicles on U.S. roads be 13 years? In which year will this occur?
7. A new car worth $36,000 is depreciating in value by $4000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $12,000.
8. A new car worth $45,000 is depreciating in value by $5000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $10,000.
9. You are choosing between two gyms. One gym offers membership for a fee of $40 plus a monthly fee of $25. The other offers membership for a fee of $15 plus a monthly fee of $30. After how many months will the total cost at each gym be the same? What will be the total cost for each gym?
10. Taxi rates are determined by local authorities. In New York City, the night-time cost of a taxi includes a base fee of $3 plus a charge of $1.56 per kilometer. In Boston, regardless of the time of day, the base fee is $2.60 with a charge of $1.75 per kilometer. For how many kilometers will the cost of a night-time taxi ride in each city be the same? Round to the nearest kilometer. What will the cost be in each city for the rounded number of kilometers?
11. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. How many times would you need to cross the bridge for the costs of the two toll options to be the same?
12. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the total cost without the pass is the same as the total cost with the pass.
13. In 2010, there were 13,300 students at college A, with a projected enrollment increase of 1000 students per year. In the same year, there were 26,800 students at college B, with a projected enrollment decline of 500 students per year. According to these projections, when did the colleges have the same enrollment? What was the enrollment in each college at that time?
14. In 2000, the population of Greece was 10,600,000, with projections of a population decrease of 28,000 people per year. In the same year, the population of Belgium was 10,200,000, with projections of a population decrease of 12,000 people per year. (Source: United Nations) According to these projections, when will the two countries have the same population? What will be the population at that time?
15. After a 20% reduction, you purchase a television for $336. What was the television’s price before the reduction?
16. After a 30% reduction, you purchase wireless earbuds for $90.30. What was the earbuds’ price before the reduction?
17. Including a 10.5% hotel tax, your room in San Diego cost $216.58 per night. Find the nightly cost before the tax was added.
18. Including a 17.4% hotel tax, your room in Chicago cost $287.63 per night. Find the nightly cost before the tax was added.
Exercises 19–20 involve markup, the amount added to the dealer’s cost of an item to arrive at the selling price of that item.
19. The selling price of a refrigerator is $1198. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the refrigerator?
20. The selling price of a scientific calculator is $15. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the calculator?
21. You invested $20,000 in two accounts paying 1.45% and 1.59% annual interest. If the total interest earned for the year was $307.50, how much was invested at each rate?
22. You invested $30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was $705.88, how much was invested at each rate?
23. Things did not go quite as planned. You invested $10,000, part of it in a stock that realized a 12% gain. However, the rest of the money suffered a 5% loss. If you had an overall gain of $520, how much was invested at each rate?
24. Things did not go quite as planned. You invested $15,000, part of it in a stock that realized a 15% gain. However, the rest of the money suffered a 7% loss. If you had an overall gain of $1590, how much was invested at each rate?
25. A rectangular soccer field is twice as long as it is wide. If the perimeter of the soccer field is 300 yards, what are its dimensions?
26. A rectangular swimming pool is three times as long as it is wide. If the perimeter of the pool is 320 feet, what are its dimensions?
27. The length of the rectangular tennis court at Wimbledon is 6 feet longer than twice the width. If the court’s perimeter is 228 feet, what are the court’s dimensions?
28. The length of a rectangular pool is 6 meters less than twice the width. If the pool’s perimeter is 126 meters, what are its dimensions?
29. The rectangular painting in the figure shown measures 12 inches by 16 inches and is surrounded by a frame of uniform width around the four edges. The perimeter of the rectangle formed by the painting and its frame is 72 inches. Determine the width of the frame.
30. The rectangular swimming pool in the figure shown measures 40 feet by 60 feet and is surrounded by a path of uniform width around the four edges. The perimeter of the rectangle formed by the pool and the surrounding path is 248 feet. Determine the width of the path.
31. The length of a rectangular sign is 3 feet longer than the width. If the sign’s area is 54 square feet, find its length and width.
32. A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.
33. Each side of a square is lengthened by 3 inches. The area of this new, larger square is 64 square inches. Find the length of a side of the original square.
34. Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 square inches. Find the length of a side of the original square.
35. A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width. If the area of the pool and the path combined is 600 square meters, what is the width of the path?
36. A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12 meters. A path of uniform width is to surround the garden. If the area of the lot is 378 square meters, find the width of the path surrounding the garden.
37. As part of a landscaping project, you put in a flower bed measuring 20 feet by 30 feet. To finish off the project, you are putting in a uniform border of pine bark around the outside of the rectangular garden. You have enough pine bark to cover 336 square feet. How wide should the border be?
38. As part of a landscaping project, you put in a flower bed measuring 10 feet by 12 feet. You plan to surround the bed with a uniform border of low-growing plants that require 1 square foot each when mature. If you have 168 of these plants, how wide a strip around the flower bed should you prepare for the border?
39. A 20-foot ladder is 15 feet from a house. How far up the house, to the nearest tenth of a foot, does the ladder reach?
40. The base of a 30-foot ladder is 10 feet from a building. If the ladder reaches the flat roof, how tall, to the nearest tenth of a foot, is the building?
41. A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is 1 foot longer than the height that it reaches on the tree. Find the length of the wire.
42. A tree is supported by a wire anchored in the ground 15 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. Find the length of the wire.
43. A rectangular piece of land whose length its twice its width has a diagonal distance of 64 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
44. A rectangular piece of land whose length is three times its width has a diagonal distance of 92 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
45. A group of people share equally in a $20,000,000 lottery. Before the money is divided, two more winning ticket holders are declared. As a result, each person’s share is reduced by $500,000. How many people were in the original group of winners?
46. A group of friends agrees to share the cost of a $480,000 vacation condominium equally. Before the purchase is made, four more people join the group and enter the agreement. As a result, each person’s share is reduced by $32,000. How many people were in the original group?
In Exercises 47–50, use the formula
47. A car can travel 300 miles in the same amount of time it takes a bus to travel 180 miles. If the average velocity of the bus is 20 miles per hour slower than the average velocity of the car, find the average velocity for each.
48. A passenger train can travel 240 miles in the same amount of time it takes a freight train to travel 160 miles. If the average velocity of the freight train is 20 miles per hour slower than the average velocity of the passenger train, find the average velocity of each.
49. You ride your bike to campus a distance of 5 miles and return home on the same route. Going to campus, you ride mostly downhill and average 9 miles per hour faster than on your return trip home. If the round trip takes one hour and ten minutes—that is hours—what is your average velocity on the return trip?
50. An engine pulls a train 140 miles. Then a second engine, whose average velocity is 5 miles per hour faster than the first engine, takes over and pulls the train 200 miles. The total time required for both engines is 9 hours. Find the average velocity of each engine.
51. An automobile repair shop charged a customer $1182, listing $357 for parts and the remainder for labor. If the cost of labor is $75 per hour, how many hours of labor did it take to repair the car?
52. A repair bill on a sailboat came to $2356, including $826 for parts and the remainder for labor. If the cost of labor is $90 per hour, how many hours of labor did it take to repair the sailboat?
53. For an international telephone call, a telephone company charges $0.43 for the first minute, $0.32 for each additional minute, and a $2.10 service charge. If the cost of a call is $5.73, how long did the person talk?
54. A job pays an annual salary of $57,900, which includes a holiday bonus of $1500. If paychecks are issued twice a month, what is the gross amount for each paycheck?
55. You have 35 hits in 140 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.30?
56. You have 30 hits in 120 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.28?
Explaining the Concepts
57. In your own words, describe a step-by-step approach for solving algebraic word problems.
58. Write an original word problem that can be solved using a linear equation. Then solve the problem.
Critical Thinking Exercises
MAKE SENSE? In Exercises 59–61, determine whether each statement makes sense or does not make sense, and explain your reasoning.
59. By modeling attitudes of college freshmen from 1969 through 2018, I can make precise predictions about the attitudes of the freshman class of 2040.
60. I find the hardest part in solving a word problem is writing the equation that models the verbal conditions.
61. After a 35% reduction, a laptop’s price is $780, so I determined the original price, by solving
62. At the north campus of a performing arts school, 10% of the students are music majors. At the south campus, 90% of the students are music majors. The campuses are merged into one east campus. If 42% of the 1000 students at the east campus are music majors, how many students did the north and south campuses have before the merger?
63. The price of a dress is reduced by 40%. When the dress still does not sell, it is reduced by 40% of the reduced price. If the price of the dress after both reductions is $72, what was the original price?
64. If I am three times your age and in 20 years I’ll be twice your age, how old are we?
65. Suppose that we agree to pay you 8¢ for every problem in this chapter that you solve correctly and fine you 5¢ for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?
66. It was wartime when Dick and Jane found out Jane was pregnant. Dick was drafted and made out a will, deciding that $14,000 in a savings account was to be divided between his wife and his child-to-be. Rather strangely, and certainly with gender bias, Dick stipulated that if the child were a boy, he would get twice the amount of the mother’s portion. If it were a girl, the mother would get twice the amount the girl was to receive. We’ll never know what Dick was thinking of, for (as fate would have it) he did not return from war. Jane gave birth to twins—a boy and a girl. How was the money divided?
67. A thief steals a number of rare plants from a nursery. On the way out, the thief meets three security guards, one after another. To each security guard, the thief is forced to give one-half the plants that he still has, plus two more. Finally, the thief leaves the nursery with one lone palm. How many plants were originally stolen?
Group Exercise
68. One of the best ways to learn how to solve a word problem in algebra is to design word problems of your own. Creating a word problem makes you very aware of precisely how much information is needed to solve the problem. You must also focus on the best way to present information to a reader and on how much information to give. As you write your problem, you gain skills that will help you solve problems created by others.
The group should design five different word problems that can be solved using linear equations. All of the problems should be on different topics. For example, the group should not have more than one problem on simple interest. The group should turn in both the problems and their algebraic solutions.
Preview Exercises
Exercises 69–71 will help you prepare for the material covered in the next section.
69. Is a solution of
70. Solve:
71. Solve:
P.8: Exercise Set
P.8 Exercise Set
Practice and Application Exercises
How will you spend your average life expectancy of 78 years? The bar graph shows the average number of years you will devote to each of your most time-consuming activities. Exercises 1–2 are based on the data displayed by the graph.
1. According to the U.S. Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 19. Over your lifetime, how many years will you spend on each of these activities?
2. According to the U.S. Bureau of Labor Statistics, you will devote 32 years to sleeping and eating. The number of years sleeping will exceed the number of years eating by 24. Over your lifetime, how many years will you spend on each of these activities?
The bar graph shows median yearly earnings of full-time workers in the United States for people 25 years and over for three occupations and two levels of education. Exercises 3–4 are based on the data displayed by the graph.
3. The median yearly salary of a general manager with a bachelor’s degree or higher is $31,000 less than twice that of a general manager with just a high school diploma. Combined, two managers with each of these educational attainments earn $149,300. Find the median yearly salary of general managers with each of these levels of education.
4. The median yearly salary of a retail salesperson with a bachelor’s degree or higher is $14,300 less than twice that of a retail salesperson with just a high school diploma. Combined, two salespeople with each of these educational attainments earn $79,900. Find the median yearly salary of salespeople with each of these levels of education.
Despite booming new car sales with their cha-ching sounds, the average age of vehicles on U.S. roads is not going down. The bar graph shows the average price of new cars in the United States and the average age of cars on U.S. roads for two selected years. Exercises 5–6 are based on the information displayed by the graph.
5. In 2019, the average price of a new car was $38,900. For the period shown, new-car prices increased by approximately $800 per year. If this trend continues, how many years after 2019 will the price of a new car average $44,500? In which year will this occur?
6. In 2019, the average age of cars on U.S. roads was 11.8 years. For the period shown, this average age increased by approximately 0.15 year per year. If this trend continues, how many years after 2019 will the average age of vehicles on U.S. roads be 13 years? In which year will this occur?
7. A new car worth $36,000 is depreciating in value by $4000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $12,000.
8. A new car worth $45,000 is depreciating in value by $5000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $10,000.
9. You are choosing between two gyms. One gym offers membership for a fee of $40 plus a monthly fee of $25. The other offers membership for a fee of $15 plus a monthly fee of $30. After how many months will the total cost at each gym be the same? What will be the total cost for each gym?
10. Taxi rates are determined by local authorities. In New York City, the night-time cost of a taxi includes a base fee of $3 plus a charge of $1.56 per kilometer. In Boston, regardless of the time of day, the base fee is $2.60 with a charge of $1.75 per kilometer. For how many kilometers will the cost of a night-time taxi ride in each city be the same? Round to the nearest kilometer. What will the cost be in each city for the rounded number of kilometers?
11. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. How many times would you need to cross the bridge for the costs of the two toll options to be the same?
12. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the total cost without the pass is the same as the total cost with the pass.
13. In 2010, there were 13,300 students at college A, with a projected enrollment increase of 1000 students per year. In the same year, there were 26,800 students at college B, with a projected enrollment decline of 500 students per year. According to these projections, when did the colleges have the same enrollment? What was the enrollment in each college at that time?
14. In 2000, the population of Greece was 10,600,000, with projections of a population decrease of 28,000 people per year. In the same year, the population of Belgium was 10,200,000, with projections of a population decrease of 12,000 people per year. (Source: United Nations) According to these projections, when will the two countries have the same population? What will be the population at that time?
15. After a 20% reduction, you purchase a television for $336. What was the television’s price before the reduction?
16. After a 30% reduction, you purchase wireless earbuds for $90.30. What was the earbuds’ price before the reduction?
17. Including a 10.5% hotel tax, your room in San Diego cost $216.58 per night. Find the nightly cost before the tax was added.
18. Including a 17.4% hotel tax, your room in Chicago cost $287.63 per night. Find the nightly cost before the tax was added.
Exercises 19–20 involve markup, the amount added to the dealer’s cost of an item to arrive at the selling price of that item.
19. The selling price of a refrigerator is $1198. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the refrigerator?
20. The selling price of a scientific calculator is $15. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the calculator?
21. You invested $20,000 in two accounts paying 1.45% and 1.59% annual interest. If the total interest earned for the year was $307.50, how much was invested at each rate?
22. You invested $30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was $705.88, how much was invested at each rate?
23. Things did not go quite as planned. You invested $10,000, part of it in a stock that realized a 12% gain. However, the rest of the money suffered a 5% loss. If you had an overall gain of $520, how much was invested at each rate?
24. Things did not go quite as planned. You invested $15,000, part of it in a stock that realized a 15% gain. However, the rest of the money suffered a 7% loss. If you had an overall gain of $1590, how much was invested at each rate?
25. A rectangular soccer field is twice as long as it is wide. If the perimeter of the soccer field is 300 yards, what are its dimensions?
26. A rectangular swimming pool is three times as long as it is wide. If the perimeter of the pool is 320 feet, what are its dimensions?
27. The length of the rectangular tennis court at Wimbledon is 6 feet longer than twice the width. If the court’s perimeter is 228 feet, what are the court’s dimensions?
28. The length of a rectangular pool is 6 meters less than twice the width. If the pool’s perimeter is 126 meters, what are its dimensions?
29. The rectangular painting in the figure shown measures 12 inches by 16 inches and is surrounded by a frame of uniform width around the four edges. The perimeter of the rectangle formed by the painting and its frame is 72 inches. Determine the width of the frame.
30. The rectangular swimming pool in the figure shown measures 40 feet by 60 feet and is surrounded by a path of uniform width around the four edges. The perimeter of the rectangle formed by the pool and the surrounding path is 248 feet. Determine the width of the path.
31. The length of a rectangular sign is 3 feet longer than the width. If the sign’s area is 54 square feet, find its length and width.
32. A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.
33. Each side of a square is lengthened by 3 inches. The area of this new, larger square is 64 square inches. Find the length of a side of the original square.
34. Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 square inches. Find the length of a side of the original square.
35. A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width. If the area of the pool and the path combined is 600 square meters, what is the width of the path?
36. A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12 meters. A path of uniform width is to surround the garden. If the area of the lot is 378 square meters, find the width of the path surrounding the garden.
37. As part of a landscaping project, you put in a flower bed measuring 20 feet by 30 feet. To finish off the project, you are putting in a uniform border of pine bark around the outside of the rectangular garden. You have enough pine bark to cover 336 square feet. How wide should the border be?
38. As part of a landscaping project, you put in a flower bed measuring 10 feet by 12 feet. You plan to surround the bed with a uniform border of low-growing plants that require 1 square foot each when mature. If you have 168 of these plants, how wide a strip around the flower bed should you prepare for the border?
39. A 20-foot ladder is 15 feet from a house. How far up the house, to the nearest tenth of a foot, does the ladder reach?
40. The base of a 30-foot ladder is 10 feet from a building. If the ladder reaches the flat roof, how tall, to the nearest tenth of a foot, is the building?
41. A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is 1 foot longer than the height that it reaches on the tree. Find the length of the wire.
42. A tree is supported by a wire anchored in the ground 15 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. Find the length of the wire.
43. A rectangular piece of land whose length its twice its width has a diagonal distance of 64 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
44. A rectangular piece of land whose length is three times its width has a diagonal distance of 92 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
45. A group of people share equally in a $20,000,000 lottery. Before the money is divided, two more winning ticket holders are declared. As a result, each person’s share is reduced by $500,000. How many people were in the original group of winners?
46. A group of friends agrees to share the cost of a $480,000 vacation condominium equally. Before the purchase is made, four more people join the group and enter the agreement. As a result, each person’s share is reduced by $32,000. How many people were in the original group?
In Exercises 47–50, use the formula
47. A car can travel 300 miles in the same amount of time it takes a bus to travel 180 miles. If the average velocity of the bus is 20 miles per hour slower than the average velocity of the car, find the average velocity for each.
48. A passenger train can travel 240 miles in the same amount of time it takes a freight train to travel 160 miles. If the average velocity of the freight train is 20 miles per hour slower than the average velocity of the passenger train, find the average velocity of each.
49. You ride your bike to campus a distance of 5 miles and return home on the same route. Going to campus, you ride mostly downhill and average 9 miles per hour faster than on your return trip home. If the round trip takes one hour and ten minutes—that is hours—what is your average velocity on the return trip?
50. An engine pulls a train 140 miles. Then a second engine, whose average velocity is 5 miles per hour faster than the first engine, takes over and pulls the train 200 miles. The total time required for both engines is 9 hours. Find the average velocity of each engine.
51. An automobile repair shop charged a customer $1182, listing $357 for parts and the remainder for labor. If the cost of labor is $75 per hour, how many hours of labor did it take to repair the car?
52. A repair bill on a sailboat came to $2356, including $826 for parts and the remainder for labor. If the cost of labor is $90 per hour, how many hours of labor did it take to repair the sailboat?
53. For an international telephone call, a telephone company charges $0.43 for the first minute, $0.32 for each additional minute, and a $2.10 service charge. If the cost of a call is $5.73, how long did the person talk?
54. A job pays an annual salary of $57,900, which includes a holiday bonus of $1500. If paychecks are issued twice a month, what is the gross amount for each paycheck?
55. You have 35 hits in 140 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.30?
56. You have 30 hits in 120 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.28?
Explaining the Concepts
57. In your own words, describe a step-by-step approach for solving algebraic word problems.
58. Write an original word problem that can be solved using a linear equation. Then solve the problem.
Critical Thinking Exercises
MAKE SENSE? In Exercises 59–61, determine whether each statement makes sense or does not make sense, and explain your reasoning.
59. By modeling attitudes of college freshmen from 1969 through 2018, I can make precise predictions about the attitudes of the freshman class of 2040.
60. I find the hardest part in solving a word problem is writing the equation that models the verbal conditions.
61. After a 35% reduction, a laptop’s price is $780, so I determined the original price, by solving
62. At the north campus of a performing arts school, 10% of the students are music majors. At the south campus, 90% of the students are music majors. The campuses are merged into one east campus. If 42% of the 1000 students at the east campus are music majors, how many students did the north and south campuses have before the merger?
63. The price of a dress is reduced by 40%. When the dress still does not sell, it is reduced by 40% of the reduced price. If the price of the dress after both reductions is $72, what was the original price?
64. If I am three times your age and in 20 years I’ll be twice your age, how old are we?
65. Suppose that we agree to pay you 8¢ for every problem in this chapter that you solve correctly and fine you 5¢ for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?
66. It was wartime when Dick and Jane found out Jane was pregnant. Dick was drafted and made out a will, deciding that $14,000 in a savings account was to be divided between his wife and his child-to-be. Rather strangely, and certainly with gender bias, Dick stipulated that if the child were a boy, he would get twice the amount of the mother’s portion. If it were a girl, the mother would get twice the amount the girl was to receive. We’ll never know what Dick was thinking of, for (as fate would have it) he did not return from war. Jane gave birth to twins—a boy and a girl. How was the money divided?
67. A thief steals a number of rare plants from a nursery. On the way out, the thief meets three security guards, one after another. To each security guard, the thief is forced to give one-half the plants that he still has, plus two more. Finally, the thief leaves the nursery with one lone palm. How many plants were originally stolen?
Group Exercise
68. One of the best ways to learn how to solve a word problem in algebra is to design word problems of your own. Creating a word problem makes you very aware of precisely how much information is needed to solve the problem. You must also focus on the best way to present information to a reader and on how much information to give. As you write your problem, you gain skills that will help you solve problems created by others.
The group should design five different word problems that can be solved using linear equations. All of the problems should be on different topics. For example, the group should not have more than one problem on simple interest. The group should turn in both the problems and their algebraic solutions.
Preview Exercises
Exercises 69–71 will help you prepare for the material covered in the next section.
69. Is a solution of
70. Solve:
71. Solve:
P.8: Exercise Set
P.8 Exercise Set
Practice and Application Exercises
How will you spend your average life expectancy of 78 years? The bar graph shows the average number of years you will devote to each of your most time-consuming activities. Exercises 1–2 are based on the data displayed by the graph.
1. According to the U.S. Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 19. Over your lifetime, how many years will you spend on each of these activities?
2. According to the U.S. Bureau of Labor Statistics, you will devote 32 years to sleeping and eating. The number of years sleeping will exceed the number of years eating by 24. Over your lifetime, how many years will you spend on each of these activities?
The bar graph shows median yearly earnings of full-time workers in the United States for people 25 years and over for three occupations and two levels of education. Exercises 3–4 are based on the data displayed by the graph.
3. The median yearly salary of a general manager with a bachelor’s degree or higher is $31,000 less than twice that of a general manager with just a high school diploma. Combined, two managers with each of these educational attainments earn $149,300. Find the median yearly salary of general managers with each of these levels of education.
4. The median yearly salary of a retail salesperson with a bachelor’s degree or higher is $14,300 less than twice that of a retail salesperson with just a high school diploma. Combined, two salespeople with each of these educational attainments earn $79,900. Find the median yearly salary of salespeople with each of these levels of education.
Despite booming new car sales with their cha-ching sounds, the average age of vehicles on U.S. roads is not going down. The bar graph shows the average price of new cars in the United States and the average age of cars on U.S. roads for two selected years. Exercises 5–6 are based on the information displayed by the graph.
5. In 2019, the average price of a new car was $38,900. For the period shown, new-car prices increased by approximately $800 per year. If this trend continues, how many years after 2019 will the price of a new car average $44,500? In which year will this occur?
6. In 2019, the average age of cars on U.S. roads was 11.8 years. For the period shown, this average age increased by approximately 0.15 year per year. If this trend continues, how many years after 2019 will the average age of vehicles on U.S. roads be 13 years? In which year will this occur?
7. A new car worth $36,000 is depreciating in value by $4000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $12,000.
8. A new car worth $45,000 is depreciating in value by $5000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $10,000.
9. You are choosing between two gyms. One gym offers membership for a fee of $40 plus a monthly fee of $25. The other offers membership for a fee of $15 plus a monthly fee of $30. After how many months will the total cost at each gym be the same? What will be the total cost for each gym?
10. Taxi rates are determined by local authorities. In New York City, the night-time cost of a taxi includes a base fee of $3 plus a charge of $1.56 per kilometer. In Boston, regardless of the time of day, the base fee is $2.60 with a charge of $1.75 per kilometer. For how many kilometers will the cost of a night-time taxi ride in each city be the same? Round to the nearest kilometer. What will the cost be in each city for the rounded number of kilometers?
11. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. How many times would you need to cross the bridge for the costs of the two toll options to be the same?
12. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the total cost without the pass is the same as the total cost with the pass.
13. In 2010, there were 13,300 students at college A, with a projected enrollment increase of 1000 students per year. In the same year, there were 26,800 students at college B, with a projected enrollment decline of 500 students per year. According to these projections, when did the colleges have the same enrollment? What was the enrollment in each college at that time?
14. In 2000, the population of Greece was 10,600,000, with projections of a population decrease of 28,000 people per year. In the same year, the population of Belgium was 10,200,000, with projections of a population decrease of 12,000 people per year. (Source: United Nations) According to these projections, when will the two countries have the same population? What will be the population at that time?
15. After a 20% reduction, you purchase a television for $336. What was the television’s price before the reduction?
16. After a 30% reduction, you purchase wireless earbuds for $90.30. What was the earbuds’ price before the reduction?
17. Including a 10.5% hotel tax, your room in San Diego cost $216.58 per night. Find the nightly cost before the tax was added.
18. Including a 17.4% hotel tax, your room in Chicago cost $287.63 per night. Find the nightly cost before the tax was added.
Exercises 19–20 involve markup, the amount added to the dealer’s cost of an item to arrive at the selling price of that item.
19. The selling price of a refrigerator is $1198. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the refrigerator?
20. The selling price of a scientific calculator is $15. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the calculator?
21. You invested $20,000 in two accounts paying 1.45% and 1.59% annual interest. If the total interest earned for the year was $307.50, how much was invested at each rate?
22. You invested $30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was $705.88, how much was invested at each rate?
23. Things did not go quite as planned. You invested $10,000, part of it in a stock that realized a 12% gain. However, the rest of the money suffered a 5% loss. If you had an overall gain of $520, how much was invested at each rate?
24. Things did not go quite as planned. You invested $15,000, part of it in a stock that realized a 15% gain. However, the rest of the money suffered a 7% loss. If you had an overall gain of $1590, how much was invested at each rate?
25. A rectangular soccer field is twice as long as it is wide. If the perimeter of the soccer field is 300 yards, what are its dimensions?
26. A rectangular swimming pool is three times as long as it is wide. If the perimeter of the pool is 320 feet, what are its dimensions?
27. The length of the rectangular tennis court at Wimbledon is 6 feet longer than twice the width. If the court’s perimeter is 228 feet, what are the court’s dimensions?
28. The length of a rectangular pool is 6 meters less than twice the width. If the pool’s perimeter is 126 meters, what are its dimensions?
29. The rectangular painting in the figure shown measures 12 inches by 16 inches and is surrounded by a frame of uniform width around the four edges. The perimeter of the rectangle formed by the painting and its frame is 72 inches. Determine the width of the frame.
30. The rectangular swimming pool in the figure shown measures 40 feet by 60 feet and is surrounded by a path of uniform width around the four edges. The perimeter of the rectangle formed by the pool and the surrounding path is 248 feet. Determine the width of the path.
31. The length of a rectangular sign is 3 feet longer than the width. If the sign’s area is 54 square feet, find its length and width.
32. A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.
33. Each side of a square is lengthened by 3 inches. The area of this new, larger square is 64 square inches. Find the length of a side of the original square.
34. Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 square inches. Find the length of a side of the original square.
35. A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width. If the area of the pool and the path combined is 600 square meters, what is the width of the path?
36. A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12 meters. A path of uniform width is to surround the garden. If the area of the lot is 378 square meters, find the width of the path surrounding the garden.
37. As part of a landscaping project, you put in a flower bed measuring 20 feet by 30 feet. To finish off the project, you are putting in a uniform border of pine bark around the outside of the rectangular garden. You have enough pine bark to cover 336 square feet. How wide should the border be?
38. As part of a landscaping project, you put in a flower bed measuring 10 feet by 12 feet. You plan to surround the bed with a uniform border of low-growing plants that require 1 square foot each when mature. If you have 168 of these plants, how wide a strip around the flower bed should you prepare for the border?
39. A 20-foot ladder is 15 feet from a house. How far up the house, to the nearest tenth of a foot, does the ladder reach?
40. The base of a 30-foot ladder is 10 feet from a building. If the ladder reaches the flat roof, how tall, to the nearest tenth of a foot, is the building?
41. A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is 1 foot longer than the height that it reaches on the tree. Find the length of the wire.
42. A tree is supported by a wire anchored in the ground 15 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. Find the length of the wire.
43. A rectangular piece of land whose length its twice its width has a diagonal distance of 64 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
44. A rectangular piece of land whose length is three times its width has a diagonal distance of 92 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
45. A group of people share equally in a $20,000,000 lottery. Before the money is divided, two more winning ticket holders are declared. As a result, each person’s share is reduced by $500,000. How many people were in the original group of winners?
46. A group of friends agrees to share the cost of a $480,000 vacation condominium equally. Before the purchase is made, four more people join the group and enter the agreement. As a result, each person’s share is reduced by $32,000. How many people were in the original group?
In Exercises 47–50, use the formula
47. A car can travel 300 miles in the same amount of time it takes a bus to travel 180 miles. If the average velocity of the bus is 20 miles per hour slower than the average velocity of the car, find the average velocity for each.
48. A passenger train can travel 240 miles in the same amount of time it takes a freight train to travel 160 miles. If the average velocity of the freight train is 20 miles per hour slower than the average velocity of the passenger train, find the average velocity of each.
49. You ride your bike to campus a distance of 5 miles and return home on the same route. Going to campus, you ride mostly downhill and average 9 miles per hour faster than on your return trip home. If the round trip takes one hour and ten minutes—that is hours—what is your average velocity on the return trip?
50. An engine pulls a train 140 miles. Then a second engine, whose average velocity is 5 miles per hour faster than the first engine, takes over and pulls the train 200 miles. The total time required for both engines is 9 hours. Find the average velocity of each engine.
51. An automobile repair shop charged a customer $1182, listing $357 for parts and the remainder for labor. If the cost of labor is $75 per hour, how many hours of labor did it take to repair the car?
52. A repair bill on a sailboat came to $2356, including $826 for parts and the remainder for labor. If the cost of labor is $90 per hour, how many hours of labor did it take to repair the sailboat?
53. For an international telephone call, a telephone company charges $0.43 for the first minute, $0.32 for each additional minute, and a $2.10 service charge. If the cost of a call is $5.73, how long did the person talk?
54. A job pays an annual salary of $57,900, which includes a holiday bonus of $1500. If paychecks are issued twice a month, what is the gross amount for each paycheck?
55. You have 35 hits in 140 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.30?
56. You have 30 hits in 120 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.28?
Explaining the Concepts
57. In your own words, describe a step-by-step approach for solving algebraic word problems.
58. Write an original word problem that can be solved using a linear equation. Then solve the problem.
Critical Thinking Exercises
MAKE SENSE? In Exercises 59–61, determine whether each statement makes sense or does not make sense, and explain your reasoning.
59. By modeling attitudes of college freshmen from 1969 through 2018, I can make precise predictions about the attitudes of the freshman class of 2040.
60. I find the hardest part in solving a word problem is writing the equation that models the verbal conditions.
61. After a 35% reduction, a laptop’s price is $780, so I determined the original price, by solving
62. At the north campus of a performing arts school, 10% of the students are music majors. At the south campus, 90% of the students are music majors. The campuses are merged into one east campus. If 42% of the 1000 students at the east campus are music majors, how many students did the north and south campuses have before the merger?
63. The price of a dress is reduced by 40%. When the dress still does not sell, it is reduced by 40% of the reduced price. If the price of the dress after both reductions is $72, what was the original price?
64. If I am three times your age and in 20 years I’ll be twice your age, how old are we?
65. Suppose that we agree to pay you 8¢ for every problem in this chapter that you solve correctly and fine you 5¢ for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?
66. It was wartime when Dick and Jane found out Jane was pregnant. Dick was drafted and made out a will, deciding that $14,000 in a savings account was to be divided between his wife and his child-to-be. Rather strangely, and certainly with gender bias, Dick stipulated that if the child were a boy, he would get twice the amount of the mother’s portion. If it were a girl, the mother would get twice the amount the girl was to receive. We’ll never know what Dick was thinking of, for (as fate would have it) he did not return from war. Jane gave birth to twins—a boy and a girl. How was the money divided?
67. A thief steals a number of rare plants from a nursery. On the way out, the thief meets three security guards, one after another. To each security guard, the thief is forced to give one-half the plants that he still has, plus two more. Finally, the thief leaves the nursery with one lone palm. How many plants were originally stolen?
Group Exercise
68. One of the best ways to learn how to solve a word problem in algebra is to design word problems of your own. Creating a word problem makes you very aware of precisely how much information is needed to solve the problem. You must also focus on the best way to present information to a reader and on how much information to give. As you write your problem, you gain skills that will help you solve problems created by others.
The group should design five different word problems that can be solved using linear equations. All of the problems should be on different topics. For example, the group should not have more than one problem on simple interest. The group should turn in both the problems and their algebraic solutions.
Preview Exercises
Exercises 69–71 will help you prepare for the material covered in the next section.
69. Is a solution of
70. Solve:
71. Solve:
P.8: Exercise Set
P.8 Exercise Set
Practice and Application Exercises
How will you spend your average life expectancy of 78 years? The bar graph shows the average number of years you will devote to each of your most time-consuming activities. Exercises 1–2 are based on the data displayed by the graph.
1. According to the U.S. Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 19. Over your lifetime, how many years will you spend on each of these activities?
2. According to the U.S. Bureau of Labor Statistics, you will devote 32 years to sleeping and eating. The number of years sleeping will exceed the number of years eating by 24. Over your lifetime, how many years will you spend on each of these activities?
The bar graph shows median yearly earnings of full-time workers in the United States for people 25 years and over for three occupations and two levels of education. Exercises 3–4 are based on the data displayed by the graph.
3. The median yearly salary of a general manager with a bachelor’s degree or higher is $31,000 less than twice that of a general manager with just a high school diploma. Combined, two managers with each of these educational attainments earn $149,300. Find the median yearly salary of general managers with each of these levels of education.
4. The median yearly salary of a retail salesperson with a bachelor’s degree or higher is $14,300 less than twice that of a retail salesperson with just a high school diploma. Combined, two salespeople with each of these educational attainments earn $79,900. Find the median yearly salary of salespeople with each of these levels of education.
Despite booming new car sales with their cha-ching sounds, the average age of vehicles on U.S. roads is not going down. The bar graph shows the average price of new cars in the United States and the average age of cars on U.S. roads for two selected years. Exercises 5–6 are based on the information displayed by the graph.
5. In 2019, the average price of a new car was $38,900. For the period shown, new-car prices increased by approximately $800 per year. If this trend continues, how many years after 2019 will the price of a new car average $44,500? In which year will this occur?
6. In 2019, the average age of cars on U.S. roads was 11.8 years. For the period shown, this average age increased by approximately 0.15 year per year. If this trend continues, how many years after 2019 will the average age of vehicles on U.S. roads be 13 years? In which year will this occur?
7. A new car worth $36,000 is depreciating in value by $4000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $12,000.
8. A new car worth $45,000 is depreciating in value by $5000 per year.
Write a formula that models the car’s value, y, in dollars, after x years.
Use the formula from part (a) to determine after how many years the car’s value will be $10,000.
9. You are choosing between two gyms. One gym offers membership for a fee of $40 plus a monthly fee of $25. The other offers membership for a fee of $15 plus a monthly fee of $30. After how many months will the total cost at each gym be the same? What will be the total cost for each gym?
10. Taxi rates are determined by local authorities. In New York City, the night-time cost of a taxi includes a base fee of $3 plus a charge of $1.56 per kilometer. In Boston, regardless of the time of day, the base fee is $2.60 with a charge of $1.75 per kilometer. For how many kilometers will the cost of a night-time taxi ride in each city be the same? Round to the nearest kilometer. What will the cost be in each city for the rounded number of kilometers?
11. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. How many times would you need to cross the bridge for the costs of the two toll options to be the same?
12. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the total cost without the pass is the same as the total cost with the pass.
13. In 2010, there were 13,300 students at college A, with a projected enrollment increase of 1000 students per year. In the same year, there were 26,800 students at college B, with a projected enrollment decline of 500 students per year. According to these projections, when did the colleges have the same enrollment? What was the enrollment in each college at that time?
14. In 2000, the population of Greece was 10,600,000, with projections of a population decrease of 28,000 people per year. In the same year, the population of Belgium was 10,200,000, with projections of a population decrease of 12,000 people per year. (Source: United Nations) According to these projections, when will the two countries have the same population? What will be the population at that time?
15. After a 20% reduction, you purchase a television for $336. What was the television’s price before the reduction?
16. After a 30% reduction, you purchase wireless earbuds for $90.30. What was the earbuds’ price before the reduction?
17. Including a 10.5% hotel tax, your room in San Diego cost $216.58 per night. Find the nightly cost before the tax was added.
18. Including a 17.4% hotel tax, your room in Chicago cost $287.63 per night. Find the nightly cost before the tax was added.
Exercises 19–20 involve markup, the amount added to the dealer’s cost of an item to arrive at the selling price of that item.
19. The selling price of a refrigerator is $1198. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the refrigerator?
20. The selling price of a scientific calculator is $15. If the markup is 25% of the dealer’s cost, what is the dealer’s cost of the calculator?
21. You invested $20,000 in two accounts paying 1.45% and 1.59% annual interest. If the total interest earned for the year was $307.50, how much was invested at each rate?
22. You invested $30,000 in two accounts paying 2.19% and 2.45% annual interest. If the total interest earned for the year was $705.88, how much was invested at each rate?
23. Things did not go quite as planned. You invested $10,000, part of it in a stock that realized a 12% gain. However, the rest of the money suffered a 5% loss. If you had an overall gain of $520, how much was invested at each rate?
24. Things did not go quite as planned. You invested $15,000, part of it in a stock that realized a 15% gain. However, the rest of the money suffered a 7% loss. If you had an overall gain of $1590, how much was invested at each rate?
25. A rectangular soccer field is twice as long as it is wide. If the perimeter of the soccer field is 300 yards, what are its dimensions?
26. A rectangular swimming pool is three times as long as it is wide. If the perimeter of the pool is 320 feet, what are its dimensions?
27. The length of the rectangular tennis court at Wimbledon is 6 feet longer than twice the width. If the court’s perimeter is 228 feet, what are the court’s dimensions?
28. The length of a rectangular pool is 6 meters less than twice the width. If the pool’s perimeter is 126 meters, what are its dimensions?
29. The rectangular painting in the figure shown measures 12 inches by 16 inches and is surrounded by a frame of uniform width around the four edges. The perimeter of the rectangle formed by the painting and its frame is 72 inches. Determine the width of the frame.
30. The rectangular swimming pool in the figure shown measures 40 feet by 60 feet and is surrounded by a path of uniform width around the four edges. The perimeter of the rectangle formed by the pool and the surrounding path is 248 feet. Determine the width of the path.
31. The length of a rectangular sign is 3 feet longer than the width. If the sign’s area is 54 square feet, find its length and width.
32. A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.
33. Each side of a square is lengthened by 3 inches. The area of this new, larger square is 64 square inches. Find the length of a side of the original square.
34. Each side of a square is lengthened by 2 inches. The area of this new, larger square is 36 square inches. Find the length of a side of the original square.
35. A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width. If the area of the pool and the path combined is 600 square meters, what is the width of the path?
36. A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12 meters. A path of uniform width is to surround the garden. If the area of the lot is 378 square meters, find the width of the path surrounding the garden.
37. As part of a landscaping project, you put in a flower bed measuring 20 feet by 30 feet. To finish off the project, you are putting in a uniform border of pine bark around the outside of the rectangular garden. You have enough pine bark to cover 336 square feet. How wide should the border be?
38. As part of a landscaping project, you put in a flower bed measuring 10 feet by 12 feet. You plan to surround the bed with a uniform border of low-growing plants that require 1 square foot each when mature. If you have 168 of these plants, how wide a strip around the flower bed should you prepare for the border?
39. A 20-foot ladder is 15 feet from a house. How far up the house, to the nearest tenth of a foot, does the ladder reach?
40. The base of a 30-foot ladder is 10 feet from a building. If the ladder reaches the flat roof, how tall, to the nearest tenth of a foot, is the building?
41. A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is 1 foot longer than the height that it reaches on the tree. Find the length of the wire.
42. A tree is supported by a wire anchored in the ground 15 feet from its base. The wire is 4 feet longer than the height that it reaches on the tree. Find the length of the wire.
43. A rectangular piece of land whose length its twice its width has a diagonal distance of 64 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
44. A rectangular piece of land whose length is three times its width has a diagonal distance of 92 yards. How many yards, to the nearest tenth of a yard, does a person save by walking diagonally across the land instead of walking its length and its width?
45. A group of people share equally in a $20,000,000 lottery. Before the money is divided, two more winning ticket holders are declared. As a result, each person’s share is reduced by $500,000. How many people were in the original group of winners?
46. A group of friends agrees to share the cost of a $480,000 vacation condominium equally. Before the purchase is made, four more people join the group and enter the agreement. As a result, each person’s share is reduced by $32,000. How many people were in the original group?
In Exercises 47–50, use the formula
47. A car can travel 300 miles in the same amount of time it takes a bus to travel 180 miles. If the average velocity of the bus is 20 miles per hour slower than the average velocity of the car, find the average velocity for each.
48. A passenger train can travel 240 miles in the same amount of time it takes a freight train to travel 160 miles. If the average velocity of the freight train is 20 miles per hour slower than the average velocity of the passenger train, find the average velocity of each.
49. You ride your bike to campus a distance of 5 miles and return home on the same route. Going to campus, you ride mostly downhill and average 9 miles per hour faster than on your return trip home. If the round trip takes one hour and ten minutes—that is hours—what is your average velocity on the return trip?
50. An engine pulls a train 140 miles. Then a second engine, whose average velocity is 5 miles per hour faster than the first engine, takes over and pulls the train 200 miles. The total time required for both engines is 9 hours. Find the average velocity of each engine.
51. An automobile repair shop charged a customer $1182, listing $357 for parts and the remainder for labor. If the cost of labor is $75 per hour, how many hours of labor did it take to repair the car?
52. A repair bill on a sailboat came to $2356, including $826 for parts and the remainder for labor. If the cost of labor is $90 per hour, how many hours of labor did it take to repair the sailboat?
53. For an international telephone call, a telephone company charges $0.43 for the first minute, $0.32 for each additional minute, and a $2.10 service charge. If the cost of a call is $5.73, how long did the person talk?
54. A job pays an annual salary of $57,900, which includes a holiday bonus of $1500. If paychecks are issued twice a month, what is the gross amount for each paycheck?
55. You have 35 hits in 140 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.30?
56. You have 30 hits in 120 times at bat. Your batting average is or 0.25. How many consecutive hits must you get to increase your batting average to 0.28?
Explaining the Concepts
57. In your own words, describe a step-by-step approach for solving algebraic word problems.
58. Write an original word problem that can be solved using a linear equation. Then solve the problem.
Critical Thinking Exercises
MAKE SENSE? In Exercises 59–61, determine whether each statement makes sense or does not make sense, and explain your reasoning.
59. By modeling attitudes of college freshmen from 1969 through 2018, I can make precise predictions about the attitudes of the freshman class of 2040.
60. I find the hardest part in solving a word problem is writing the equation that models the verbal conditions.
61. After a 35% reduction, a laptop’s price is $780, so I determined the original price, by solving
62. At the north campus of a performing arts school, 10% of the students are music majors. At the south campus, 90% of the students are music majors. The campuses are merged into one east campus. If 42% of the 1000 students at the east campus are music majors, how many students did the north and south campuses have before the merger?
63. The price of a dress is reduced by 40%. When the dress still does not sell, it is reduced by 40% of the reduced price. If the price of the dress after both reductions is $72, what was the original price?
64. If I am three times your age and in 20 years I’ll be twice your age, how old are we?
65. Suppose that we agree to pay you 8¢ for every problem in this chapter that you solve correctly and fine you 5¢ for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?
66. It was wartime when Dick and Jane found out Jane was pregnant. Dick was drafted and made out a will, deciding that $14,000 in a savings account was to be divided between his wife and his child-to-be. Rather strangely, and certainly with gender bias, Dick stipulated that if the child were a boy, he would get twice the amount of the mother’s portion. If it were a girl, the mother would get twice the amount the girl was to receive. We’ll never know what Dick was thinking of, for (as fate would have it) he did not return from war. Jane gave birth to twins—a boy and a girl. How was the money divided?
67. A thief steals a number of rare plants from a nursery. On the way out, the thief meets three security guards, one after another. To each security guard, the thief is forced to give one-half the plants that he still has, plus two more. Finally, the thief leaves the nursery with one lone palm. How many plants were originally stolen?
Group Exercise
68. One of the best ways to learn how to solve a word problem in algebra is to design word problems of your own. Creating a word problem makes you very aware of precisely how much information is needed to solve the problem. You must also focus on the best way to present information to a reader and on how much information to give. As you write your problem, you gain skills that will help you solve problems created by others.
The group should design five different word problems that can be solved using linear equations. All of the problems should be on different topics. For example, the group should not have more than one problem on simple interest. The group should turn in both the problems and their algebraic solutions.
Preview Exercises
Exercises 69–71 will help you prepare for the material covered in the next section.
69. Is a solution of
70. Solve:
71. Solve:
Section P.9: Linear Inequalities and Absolute Value Inequalities
Section P.9 Linear Inequalities and Absolute Value Inequalities
Rent-a-Heap, a car rental company, charges $125 per week plus $0.20 per mile to rent one of their cars. Suppose you are limited by how much money you can spend for the week: You can spend at most $335. If we let x represent the number of miles you drive the heap in a week, we can write an inequality that models the given conditions.
Placing an inequality symbol between a polynomial of degree 1 and a constant results in a linear inequality in one variable. In this section, we will study how to solve linear inequalities such as . Solving an inequality is the process of finding the set of numbers that make the inequality a true statement. These numbers are called the solutions of the inequality and we say that they satisfy the inequality. The set of all solutions is called the solution set of the inequality. Set-builder notation and a new notation, called interval notation, are used to represent these solution sets. We begin this section by looking at interval notation.
Objective 1: Use interval notation
Interval Notation
Objective 1Use interval notation.
Watch Video
Some sets of real numbers can be represented using interval notation. Suppose that a and b are two real numbers such that .
Parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpoints that are included in an interval. Parentheses are always used with ∞ or .
Table P.7 lists nine possible types of intervals used to describe subsets of real numbers.
Express each interval in set-builder notation and graph:
.
Objective 2: Find intersections and unions of intervals
Intersections and Unions of Intervals
Objective 2Find intersections and unions of intervals.
Watch Video
In Section P.1, we learned how to find intersections and unions of sets. Recall that (A intersection B) is the set of elements common to both set A and set B. By contrast, (A union B) is the set of elements in set A or in set B or in both sets.
Because intervals represent sets, it is possible to find their intersections and unions. Graphs are helpful in this process.
Finding Intersections and Unions of Two Intervals
Graph each interval on a number line.
To find the intersection, take the portion of the number line that the two graphs have in common.
To find the union, take the portion of the number line representing the total collection of numbers in the two graphs.
Example 2 Finding Intersections and Unions of Intervals
Use graphs to find each set:
.
Solution
, the intersection of the intervals and , consists of the numbers that are in both intervals.
We know that a linear equation in x can be expressed as . A linear inequality in x can be written in one of the following forms:
In each form, .
Back to our question that opened this section: How many miles can you drive your Rent-a-Heap car if you can spend at most $335? We answer the question by solving
for x. The solution procedure is nearly identical to that for solving
Our goal is to get x by itself on the left side. We do this by subtracting 125 from both sides to isolate 0.20x:
Finally, we isolate x from 0.20x by dividing both sides of the inequality by 0.20:
With at most $335 to spend, you can travel at most 1050 miles.
We started with the inequality and obtained the inequality in the final step. These inequalities have the same solution set, namely, . Inequalities such as these, with the same solution set, are said to be equivalent.
We isolated x from 0.20x by dividing both sides of by 0.20, a positive number. Let’s see what happens if we divide both sides of an inequality by a negative number. Consider the inequality . Divide 10 and 14 by
Because lies to the right of on the number line, is greater than
Notice that the direction of the inequality symbol is reversed:
In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed. When we reverse the direction of the inequality symbol, we say that we change the sense of the inequality.
We can isolate a variable in a linear inequality in the same way we isolate a variable in a linear equation. The properties on the next page are used to create equivalent inequalities.
Properties of Inequalities
Property
The Property in Words
Example
The Addition Property of Inequality
If , then .
If , then .
If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one.
Subtract 3:
Simplify:
The Positive Multiplication Property of Inequality
If and c is positive, then .
If and c is positive, then .
If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one.
Divide by 2:
Simplify:
The Negative Multiplication Property of Inequality
If and c is negative, then .
If and c is negative, then .
If we multiply or divide both sides of an inequality by the same negative quantity and reverse the direction of the inequality symbol, the resulting inequality is equivalent to the original one.
Divide by and change the sense of the inequality:
Simplify:
Example 3 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
The solution set of , or equivalently consists of all real numbers that are greater than or equal to , expressed as in set-builder notation. The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Example 4 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
Step 1 SIMPLIFY EACH SIDE. Because each side is already simplified, we can skip this step.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. We isolate the variable, x, by dividing both sides by . Because we are dividing by a negative number, we must reverse the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than , expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
If an inequality contains fractions with constants in the denominators, begin by multiplying both sides by the least common denominator. This will clear the inequality of fractions.
Example 5 Solving a Linear Inequality Containing Fractions
Solve and graph the solution set on a number line:
Solution
The denominators are 4, 3, and 4. The least common denominator is 12. We begin by multiplying both sides of the inequality by 12.
Now that the fractions have been cleared, we follow the four steps that we used in the previous example.
Step 1 SIMPLIFY EACH SIDE.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. To isolate x, we must eliminate the negative sign in front of the x. Because means , we can do this by multiplying (or dividing) both sides of the inequality by . We are multiplying by a negative number. Thus, we must reverse the direction of the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than or equal to 14, expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Objective 3: Solve linear inequalities
Solving Linear Inequalities in One Variable
Objective 3Solve linear inequalities.
Watch Video
We know that a linear equation in x can be expressed as . A linear inequality in x can be written in one of the following forms:
In each form, .
Back to our question that opened this section: How many miles can you drive your Rent-a-Heap car if you can spend at most $335? We answer the question by solving
for x. The solution procedure is nearly identical to that for solving
Our goal is to get x by itself on the left side. We do this by subtracting 125 from both sides to isolate 0.20x:
Finally, we isolate x from 0.20x by dividing both sides of the inequality by 0.20:
With at most $335 to spend, you can travel at most 1050 miles.
We started with the inequality and obtained the inequality in the final step. These inequalities have the same solution set, namely, . Inequalities such as these, with the same solution set, are said to be equivalent.
We isolated x from 0.20x by dividing both sides of by 0.20, a positive number. Let’s see what happens if we divide both sides of an inequality by a negative number. Consider the inequality . Divide 10 and 14 by
Because lies to the right of on the number line, is greater than
Notice that the direction of the inequality symbol is reversed:
In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed. When we reverse the direction of the inequality symbol, we say that we change the sense of the inequality.
We can isolate a variable in a linear inequality in the same way we isolate a variable in a linear equation. The properties on the next page are used to create equivalent inequalities.
Properties of Inequalities
Property
The Property in Words
Example
The Addition Property of Inequality
If , then .
If , then .
If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one.
Subtract 3:
Simplify:
The Positive Multiplication Property of Inequality
If and c is positive, then .
If and c is positive, then .
If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one.
Divide by 2:
Simplify:
The Negative Multiplication Property of Inequality
If and c is negative, then .
If and c is negative, then .
If we multiply or divide both sides of an inequality by the same negative quantity and reverse the direction of the inequality symbol, the resulting inequality is equivalent to the original one.
Divide by and change the sense of the inequality:
Simplify:
Example 3 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
The solution set of , or equivalently consists of all real numbers that are greater than or equal to , expressed as in set-builder notation. The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Example 4 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
Step 1 SIMPLIFY EACH SIDE. Because each side is already simplified, we can skip this step.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. We isolate the variable, x, by dividing both sides by . Because we are dividing by a negative number, we must reverse the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than , expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
If an inequality contains fractions with constants in the denominators, begin by multiplying both sides by the least common denominator. This will clear the inequality of fractions.
Example 5 Solving a Linear Inequality Containing Fractions
Solve and graph the solution set on a number line:
Solution
The denominators are 4, 3, and 4. The least common denominator is 12. We begin by multiplying both sides of the inequality by 12.
Now that the fractions have been cleared, we follow the four steps that we used in the previous example.
Step 1 SIMPLIFY EACH SIDE.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. To isolate x, we must eliminate the negative sign in front of the x. Because means , we can do this by multiplying (or dividing) both sides of the inequality by . We are multiplying by a negative number. Thus, we must reverse the direction of the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than or equal to 14, expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Objective 3: Solve linear inequalities
Solving Linear Inequalities in One Variable
Objective 3Solve linear inequalities.
Watch Video
We know that a linear equation in x can be expressed as . A linear inequality in x can be written in one of the following forms:
In each form, .
Back to our question that opened this section: How many miles can you drive your Rent-a-Heap car if you can spend at most $335? We answer the question by solving
for x. The solution procedure is nearly identical to that for solving
Our goal is to get x by itself on the left side. We do this by subtracting 125 from both sides to isolate 0.20x:
Finally, we isolate x from 0.20x by dividing both sides of the inequality by 0.20:
With at most $335 to spend, you can travel at most 1050 miles.
We started with the inequality and obtained the inequality in the final step. These inequalities have the same solution set, namely, . Inequalities such as these, with the same solution set, are said to be equivalent.
We isolated x from 0.20x by dividing both sides of by 0.20, a positive number. Let’s see what happens if we divide both sides of an inequality by a negative number. Consider the inequality . Divide 10 and 14 by
Because lies to the right of on the number line, is greater than
Notice that the direction of the inequality symbol is reversed:
In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed. When we reverse the direction of the inequality symbol, we say that we change the sense of the inequality.
We can isolate a variable in a linear inequality in the same way we isolate a variable in a linear equation. The properties on the next page are used to create equivalent inequalities.
Properties of Inequalities
Property
The Property in Words
Example
The Addition Property of Inequality
If , then .
If , then .
If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one.
Subtract 3:
Simplify:
The Positive Multiplication Property of Inequality
If and c is positive, then .
If and c is positive, then .
If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one.
Divide by 2:
Simplify:
The Negative Multiplication Property of Inequality
If and c is negative, then .
If and c is negative, then .
If we multiply or divide both sides of an inequality by the same negative quantity and reverse the direction of the inequality symbol, the resulting inequality is equivalent to the original one.
Divide by and change the sense of the inequality:
Simplify:
Example 3 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
The solution set of , or equivalently consists of all real numbers that are greater than or equal to , expressed as in set-builder notation. The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Example 4 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
Step 1 SIMPLIFY EACH SIDE. Because each side is already simplified, we can skip this step.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. We isolate the variable, x, by dividing both sides by . Because we are dividing by a negative number, we must reverse the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than , expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
If an inequality contains fractions with constants in the denominators, begin by multiplying both sides by the least common denominator. This will clear the inequality of fractions.
Example 5 Solving a Linear Inequality Containing Fractions
Solve and graph the solution set on a number line:
Solution
The denominators are 4, 3, and 4. The least common denominator is 12. We begin by multiplying both sides of the inequality by 12.
Now that the fractions have been cleared, we follow the four steps that we used in the previous example.
Step 1 SIMPLIFY EACH SIDE.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. To isolate x, we must eliminate the negative sign in front of the x. Because means , we can do this by multiplying (or dividing) both sides of the inequality by . We are multiplying by a negative number. Thus, we must reverse the direction of the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than or equal to 14, expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Objective 3: Solve linear inequalities
Solving Linear Inequalities in One Variable
Objective 3Solve linear inequalities.
Watch Video
We know that a linear equation in x can be expressed as . A linear inequality in x can be written in one of the following forms:
In each form, .
Back to our question that opened this section: How many miles can you drive your Rent-a-Heap car if you can spend at most $335? We answer the question by solving
for x. The solution procedure is nearly identical to that for solving
Our goal is to get x by itself on the left side. We do this by subtracting 125 from both sides to isolate 0.20x:
Finally, we isolate x from 0.20x by dividing both sides of the inequality by 0.20:
With at most $335 to spend, you can travel at most 1050 miles.
We started with the inequality and obtained the inequality in the final step. These inequalities have the same solution set, namely, . Inequalities such as these, with the same solution set, are said to be equivalent.
We isolated x from 0.20x by dividing both sides of by 0.20, a positive number. Let’s see what happens if we divide both sides of an inequality by a negative number. Consider the inequality . Divide 10 and 14 by
Because lies to the right of on the number line, is greater than
Notice that the direction of the inequality symbol is reversed:
In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed. When we reverse the direction of the inequality symbol, we say that we change the sense of the inequality.
We can isolate a variable in a linear inequality in the same way we isolate a variable in a linear equation. The properties on the next page are used to create equivalent inequalities.
Properties of Inequalities
Property
The Property in Words
Example
The Addition Property of Inequality
If , then .
If , then .
If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one.
Subtract 3:
Simplify:
The Positive Multiplication Property of Inequality
If and c is positive, then .
If and c is positive, then .
If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one.
Divide by 2:
Simplify:
The Negative Multiplication Property of Inequality
If and c is negative, then .
If and c is negative, then .
If we multiply or divide both sides of an inequality by the same negative quantity and reverse the direction of the inequality symbol, the resulting inequality is equivalent to the original one.
Divide by and change the sense of the inequality:
Simplify:
Example 3 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
The solution set of , or equivalently consists of all real numbers that are greater than or equal to , expressed as in set-builder notation. The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Example 4 Solving a Linear Inequality
Solve and graph the solution set on a number line:
Solution
Step 1 SIMPLIFY EACH SIDE. Because each side is already simplified, we can skip this step.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. We isolate the variable, x, by dividing both sides by . Because we are dividing by a negative number, we must reverse the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than , expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
If an inequality contains fractions with constants in the denominators, begin by multiplying both sides by the least common denominator. This will clear the inequality of fractions.
Example 5 Solving a Linear Inequality Containing Fractions
Solve and graph the solution set on a number line:
Solution
The denominators are 4, 3, and 4. The least common denominator is 12. We begin by multiplying both sides of the inequality by 12.
Now that the fractions have been cleared, we follow the four steps that we used in the previous example.
Step 1 SIMPLIFY EACH SIDE.
Step 2 COLLECT VARIABLE TERMS ON ONE SIDE AND CONSTANT TERMS ON THE OTHER SIDE. We will collect variable terms of on the left and constant terms on the right.
Step 3 ISOLATE THE VARIABLE AND SOLVE. To isolate x, we must eliminate the negative sign in front of the x. Because means , we can do this by multiplying (or dividing) both sides of the inequality by . We are multiplying by a negative number. Thus, we must reverse the direction of the inequality symbol.
Step 4 EXPRESS THE SOLUTION SET IN SET-BUILDER OR INTERVAL NOTATION AND GRAPH THE SET ON A NUMBER LINE. The solution set consists of all real numbers that are less than or equal to 14, expressed in set-builder notation as . The interval notation for this solution set is . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line:
Objective 4: Solve compound inequalities
Solving Compound Inequalities
Objective 4Solve compound inequalities.
Watch Video
We now consider two inequalities such as
expressed as a compound inequality
The word and does not appear when the inequality is written in the shorter form, although intersection is implied. The shorter form enables us to solve both inequalities at once. By performing each operation on all three parts of the inequality, our goal is to isolate x in the middle.
Example 6 Solving a Compound Inequality
Solve and graph the solution set on a number line:
Solution
We would like to isolate x in the middle. We can do this by first subtracting 1 from all three parts of the compound inequality. Then we isolate x from 2x by dividing all three parts of the inequality by 2.
The solution set consists of all real numbers greater than and less than or equal to 1, represented by in set-builder notation and in interval notation. The graph is shown as follows:
Solve and graph the solution set on a number line: .
Objective 5: Solve absolute value inequalities
Solving Inequalities with Absolute Value
Objective 5Solve absolute value inequalities.
Watch Video
We know that describes the distance of x from zero on a real number line. We can use this geometric interpretation to solve an inequality such as
This means that the distance of x from 0 is less than 2, as shown in Figure P.22. The interval shows values of x that lie less than 2 units from 0. Thus, x can lie between and 2. That is, x is greater than and less than 2. We write or .
Figure P.22 , so .
Some absolute value inequalities use the “greater than” symbol. For example, means that the distance of x from 0 is greater than 2, as shown in Figure P.23. Thus, x can be less than or greater than 2. We write or . This can be expressed in interval notation as .
Figure P.23 , so or .
These observations suggest the following principles for solving inequalities with absolute value.
Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive number,
The solutions of are the numbers that satisfy .
The solutions of are the numbers that satisfy or .
These rules are valid if < is replaced by ≤ and > is replaced by ≥.
Example 7 Solving an Absolute Value Inequality
Solve and graph the solution set on a number line: .
Solution
We rewrite the inequality without absolute value bars.
We solve the compound inequality by adding 4 to all three parts.
The solution set of consists of all real numbers greater than 1 and less than 7, denoted by or . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
Example 9 Solving an Absolute Value Inequality
Solve and graph the solution set on a number line: .
Solution
We begin by expressing the inequality with the absolute value expression on the left side:
We rewrite this inequality without absolute value bars.
Because means or , we solve and separately. Then we take the union of their solution sets.
The solution set consists of all numbers that are less than or greater than 6. The solution set is , or, in interval notation . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
Applications
In Section P.8, we solved equations to determine when two different pricing options would result in the same cost. With inequalities, we can look at the same situations and ask when one of the pricing options results in a lower cost than the other.
Our next example shows how to use an inequality to select the better deal between our two options for paying the bridge toll from Example 3 in Section P.8. We use our strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality.
Example 10 Selecting the Better Deal
You still have two options for paying the toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 and a $3 administrative fee each time you cross the bridge, for a total of $7.25 for each crossing. Find the number of times you would need to cross the bridge for the transponder option to be the better deal.
Solution
Step 1 LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are looking for the number of times you must cross the bridge to make the transponder option the better deal. Thus,
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. We are not asked to find another quantity, so we can skip this step.
Step 3 WRITE AN INEQUALITY IN x THAT MODELS THE CONDITIONS. The transponder is a better deal than toll-by-plate if the total cost with the transponder is less than the total cost of toll-by-plate.
Step 4 SOLVE THE INEQUALITY AND ANSWER THE QUESTION.
Thus, crossing the bridge more than five times makes the transponder option the better deal.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. One way to do this is to take a number of crossings greater than five and see if the transponder option is the better deal. Suppose that you cross the bridge six times.
The cost with the transponder is lower, making this option the better deal.
You drive up to a toll plaza and find booths with attendants, and you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge to make the decal option the better deal.
Objective 5: Solve absolute value inequalities
Solving Inequalities with Absolute Value
Objective 5Solve absolute value inequalities.
Watch Video
We know that describes the distance of x from zero on a real number line. We can use this geometric interpretation to solve an inequality such as
This means that the distance of x from 0 is less than 2, as shown in Figure P.22. The interval shows values of x that lie less than 2 units from 0. Thus, x can lie between and 2. That is, x is greater than and less than 2. We write or .
Figure P.22 , so .
Some absolute value inequalities use the “greater than” symbol. For example, means that the distance of x from 0 is greater than 2, as shown in Figure P.23. Thus, x can be less than or greater than 2. We write or . This can be expressed in interval notation as .
Figure P.23 , so or .
These observations suggest the following principles for solving inequalities with absolute value.
Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive number,
The solutions of are the numbers that satisfy .
The solutions of are the numbers that satisfy or .
These rules are valid if < is replaced by ≤ and > is replaced by ≥.
Example 7 Solving an Absolute Value Inequality
Solve and graph the solution set on a number line: .
Solution
We rewrite the inequality without absolute value bars.
We solve the compound inequality by adding 4 to all three parts.
The solution set of consists of all real numbers greater than 1 and less than 7, denoted by or . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
Example 9 Solving an Absolute Value Inequality
Solve and graph the solution set on a number line: .
Solution
We begin by expressing the inequality with the absolute value expression on the left side:
We rewrite this inequality without absolute value bars.
Because means or , we solve and separately. Then we take the union of their solution sets.
The solution set consists of all numbers that are less than or greater than 6. The solution set is , or, in interval notation . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
Applications
In Section P.8, we solved equations to determine when two different pricing options would result in the same cost. With inequalities, we can look at the same situations and ask when one of the pricing options results in a lower cost than the other.
Our next example shows how to use an inequality to select the better deal between our two options for paying the bridge toll from Example 3 in Section P.8. We use our strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality.
Example 10 Selecting the Better Deal
You still have two options for paying the toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 and a $3 administrative fee each time you cross the bridge, for a total of $7.25 for each crossing. Find the number of times you would need to cross the bridge for the transponder option to be the better deal.
Solution
Step 1 LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are looking for the number of times you must cross the bridge to make the transponder option the better deal. Thus,
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. We are not asked to find another quantity, so we can skip this step.
Step 3 WRITE AN INEQUALITY IN x THAT MODELS THE CONDITIONS. The transponder is a better deal than toll-by-plate if the total cost with the transponder is less than the total cost of toll-by-plate.
Step 4 SOLVE THE INEQUALITY AND ANSWER THE QUESTION.
Thus, crossing the bridge more than five times makes the transponder option the better deal.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. One way to do this is to take a number of crossings greater than five and see if the transponder option is the better deal. Suppose that you cross the bridge six times.
The cost with the transponder is lower, making this option the better deal.
You drive up to a toll plaza and find booths with attendants, and you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge to make the decal option the better deal.
Objective 5: Solve absolute value inequalities
Solving Inequalities with Absolute Value
Objective 5Solve absolute value inequalities.
Watch Video
We know that describes the distance of x from zero on a real number line. We can use this geometric interpretation to solve an inequality such as
This means that the distance of x from 0 is less than 2, as shown in Figure P.22. The interval shows values of x that lie less than 2 units from 0. Thus, x can lie between and 2. That is, x is greater than and less than 2. We write or .
Figure P.22 , so .
Some absolute value inequalities use the “greater than” symbol. For example, means that the distance of x from 0 is greater than 2, as shown in Figure P.23. Thus, x can be less than or greater than 2. We write or . This can be expressed in interval notation as .
Figure P.23 , so or .
These observations suggest the following principles for solving inequalities with absolute value.
Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive number,
The solutions of are the numbers that satisfy .
The solutions of are the numbers that satisfy or .
These rules are valid if < is replaced by ≤ and > is replaced by ≥.
Example 7 Solving an Absolute Value Inequality
Solve and graph the solution set on a number line: .
Solution
We rewrite the inequality without absolute value bars.
We solve the compound inequality by adding 4 to all three parts.
The solution set of consists of all real numbers greater than 1 and less than 7, denoted by or . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
Example 9 Solving an Absolute Value Inequality
Solve and graph the solution set on a number line: .
Solution
We begin by expressing the inequality with the absolute value expression on the left side:
We rewrite this inequality without absolute value bars.
Because means or , we solve and separately. Then we take the union of their solution sets.
The solution set consists of all numbers that are less than or greater than 6. The solution set is , or, in interval notation . The graph of the solution set is shown as follows:
Solve and graph the solution set on a number line: .
Applications
In Section P.8, we solved equations to determine when two different pricing options would result in the same cost. With inequalities, we can look at the same situations and ask when one of the pricing options results in a lower cost than the other.
Our next example shows how to use an inequality to select the better deal between our two options for paying the bridge toll from Example 3 in Section P.8. We use our strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality.
Example 10 Selecting the Better Deal
You still have two options for paying the toll so that you can get to the beach. The first option requires purchasing a transponder for $20; with the transponder, you pay a reduced toll of $3.25 each time you cross the bridge. Your second option is toll-by-plate; with this option, you pay the full toll of $4.25 and a $3 administrative fee each time you cross the bridge, for a total of $7.25 for each crossing. Find the number of times you would need to cross the bridge for the transponder option to be the better deal.
Solution
Step 1 LET x REPRESENT ONE OF THE UNKNOWN QUANTITIES. We are looking for the number of times you must cross the bridge to make the transponder option the better deal. Thus,
Step 2 REPRESENT OTHER UNKNOWN QUANTITIES IN TERMS OF x. We are not asked to find another quantity, so we can skip this step.
Step 3 WRITE AN INEQUALITY IN x THAT MODELS THE CONDITIONS. The transponder is a better deal than toll-by-plate if the total cost with the transponder is less than the total cost of toll-by-plate.
Step 4 SOLVE THE INEQUALITY AND ANSWER THE QUESTION.
Thus, crossing the bridge more than five times makes the transponder option the better deal.
Step 5 CHECK THE PROPOSED SOLUTION IN THE ORIGINAL WORDING OF THE PROBLEM. One way to do this is to take a number of crossings greater than five and see if the transponder option is the better deal. Suppose that you cross the bridge six times.
The cost with the transponder is lower, making this option the better deal.
You drive up to a toll plaza and find booths with attendants, and you can pay the toll by cash or credit card. With this option, the toll is $5 each time you cross the bridge. The attendant gives you the option of buying a bar-coded decal for $25; with the decal, you get 25% off the normal toll of $5 for each crossing. Find the number of times you would need to cross the bridge to make the decal option the better deal.
P.9: Exercise Set
P.9 Exercise Set
Practice Exercises
In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
In Exercises 15–26, use graphs to find each set.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In all exercises, use interval notation to express solution sets and graph each solution set on a number line.
In Exercises 27–48, solve each linear inequality.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–56, solve each compound inequality.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–92, solve each absolute value inequality.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–100, use interval notation to represent all values of x satisfying the given conditions.
93. and
94. and
95. and is at least 4.
96. and is at most 0.
97. and
98. and
99. and is at most 4.
100. and is at least 6.
101. When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.
102. When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.
Application Exercises
The graphs show that the three components of love, namely, passion, intimacy, and commitment, progress differently over time. Passion peaks early in a relationship and then declines. By contrast, intimacy and commitment build gradually. Use the graphs to solve Exercises 103–110.
Source: R. J. Sternberg. A Triangular Theory of Love, Psychological Review, 93, 119–135
103. Use interval notation to write an inequality that expresses for which years in a relationship intimacy is greater than commitment.
104. Use interval notation to write an inequality that expresses for which years in a relationship passion is greater than or equal to intimacy.
105. What is the relationship between passion and intimacy on the interval [5, 7)?
106. What is the relationship between intimacy and commitment on the interval [4, 7)?
107. What is the relationship between passion and commitment for
108. What is the relationship between passion and commitment for
109. What is the maximum level of intensity for passion? After how many years in a relationship does this occur?
110. After approximately how many years do levels of intensity for commitment exceed the maximum level of intensity for passion?
In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and 2012. Also shown is the percentage of households in which a person of faith is married to someone with no religion.
Source: General Social Survey, University of Chicago
models the percentage of U.S. households with an interfaith marriage, I, x years after 1988. The formula
models the percentage of U.S. households in which a person of faith is married to someone with no religion, N, x years after 1988.
Use these models to solve Exercises 111–112.
111.
In which years will more than 33% of U.S. households have an interfaith marriage?
In which years will more than 14% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage and more than 14% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage or more than 14% have a faith/no religion marriage?
112.
In which years will more than 34% of U.S. households have an interfaith marriage?
In which years will more than 15% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage and more than 15% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage or more than 15% have a faith/no religion marriage?
113. The formula for converting Fahrenheit temperature, to Celsius temperature, is
If Celsius temperature ranges from to , inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.
114. The formula for converting Celsius temperature, C, to Fahrenheit temperature, F, is
If Fahrenheit temperature ranges from 41° to 50°, inclusive, what is the range for Celsius temperature? Use interval notation to express this range.
115. If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if the number of outcomes that result in heads, satisfies Describe the number of outcomes that determine an unfair coin that is tossed 100 times.
In Exercises 116–127, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality.
116. You see a sign on a small moving truck that reads “Rent me for $20 a day.*” Oh, there’s an asterisk: There is an additional charge of $0.80 per mile. When you go online to reserve the truck, you are offered an unlimited mileage option for $60 a day. How many miles would you have to drive the truck in a day to make the unlimited mileage option the better deal?
117. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. For how many bridge crossings is toll-by-plate the better option?
118. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the electronic pass option is the better deal.
119. A city commission has proposed two tax bills. The first bill requires that a homeowner pay $1200 plus 0.5% of the assessed home value in taxes. The second bill requires taxes of $300 plus 0.9% of the assessed home value. What price range of home assessment would make the first bill a better deal?
120. A company designs and sells greeting cards. The weekly fixed cost is $10,000 and it costs $0.40 to create each card. The selling price is $2.00 per card. How many greeting cards must be designed and sold each week for the company to generate a profit?
121. A company manufactures and sells personalized stationery. The weekly fixed cost is $3000 and it costs $3.00 to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?
122. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
123. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
124. To earn an A in a course, you must have a final average of at least 90%. On the first four examinations, you have grades of 86%, 88%, 92%, and 84%. If the final examination counts as two grades, what must you get on the final to earn an A in the course?
125. On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90.
What must you get on the final to earn an A in the course?
By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than 80, you will lose your B in the course. Describe the grades on the final that will cause this to happen.
126. Parts for an automobile repair cost $254. The mechanic charges $65 per hour. If you receive an estimate for at least $351.50 and at most $481.50 for fixing the car, what is the time interval that the mechanic will be working on the job?
127. The toll to a bridge is $3.00. A three-month pass costs $7.50 and reduces the toll to $0.50. A six-month pass costs $30 and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?
Explaining the Concepts
128. When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?
129. Describe ways in which solving a linear inequality is similar to solving a linear equation.
130. Describe ways in which solving a linear inequality is different from solving a linear equation.
131. What is a compound inequality and how is it solved?
132. Describe how to solve an absolute value inequality involving the symbol Give an example.
133. Describe how to solve an absolute value inequality involving the symbol Give an example.
134. Explain why has no solution.
135. Describe the solution set of
Critical Thinking Exercises
Make Sense? In Exercises 136–139, determine whether each statement makes sense or does not make sense, and explain your reasoning.
136. I prefer interval notation over set-builder notation because it takes less space to write solution sets.
137. I can check inequalities by substituting 0 for the variable: When 0 belongs to the solution set, I should obtain a true statement, and when 0 does not belong to the solution set, I should obtain a false statement.
138. In an inequality such as I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.
139. I’ll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, is modeled by
In Exercises 140–143, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
140.
141.
142. The inequality is equivalent to
143. All irrational numbers satisfy
144. What’s wrong with this argument? Suppose x and represent two real numbers, where
The final inequality, , is impossible because we were initially given .
145. Write an absolute value inequality for which the interval shown is the solution.
146. Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without electronic passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.
Preview Exercises
Exercises 147–149 will help you prepare for the material covered in the first section of the next chapter.
147. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
148. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
149. If , find the value of y that corresponds to values of x for each integer starting with and ending with 2.
P.9: Exercise Set
P.9 Exercise Set
Practice Exercises
In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
In Exercises 15–26, use graphs to find each set.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In all exercises, use interval notation to express solution sets and graph each solution set on a number line.
In Exercises 27–48, solve each linear inequality.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–56, solve each compound inequality.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–92, solve each absolute value inequality.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–100, use interval notation to represent all values of x satisfying the given conditions.
93. and
94. and
95. and is at least 4.
96. and is at most 0.
97. and
98. and
99. and is at most 4.
100. and is at least 6.
101. When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.
102. When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.
Application Exercises
The graphs show that the three components of love, namely, passion, intimacy, and commitment, progress differently over time. Passion peaks early in a relationship and then declines. By contrast, intimacy and commitment build gradually. Use the graphs to solve Exercises 103–110.
Source: R. J. Sternberg. A Triangular Theory of Love, Psychological Review, 93, 119–135
103. Use interval notation to write an inequality that expresses for which years in a relationship intimacy is greater than commitment.
104. Use interval notation to write an inequality that expresses for which years in a relationship passion is greater than or equal to intimacy.
105. What is the relationship between passion and intimacy on the interval [5, 7)?
106. What is the relationship between intimacy and commitment on the interval [4, 7)?
107. What is the relationship between passion and commitment for
108. What is the relationship between passion and commitment for
109. What is the maximum level of intensity for passion? After how many years in a relationship does this occur?
110. After approximately how many years do levels of intensity for commitment exceed the maximum level of intensity for passion?
In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and 2012. Also shown is the percentage of households in which a person of faith is married to someone with no religion.
Source: General Social Survey, University of Chicago
models the percentage of U.S. households with an interfaith marriage, I, x years after 1988. The formula
models the percentage of U.S. households in which a person of faith is married to someone with no religion, N, x years after 1988.
Use these models to solve Exercises 111–112.
111.
In which years will more than 33% of U.S. households have an interfaith marriage?
In which years will more than 14% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage and more than 14% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage or more than 14% have a faith/no religion marriage?
112.
In which years will more than 34% of U.S. households have an interfaith marriage?
In which years will more than 15% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage and more than 15% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage or more than 15% have a faith/no religion marriage?
113. The formula for converting Fahrenheit temperature, to Celsius temperature, is
If Celsius temperature ranges from to , inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.
114. The formula for converting Celsius temperature, C, to Fahrenheit temperature, F, is
If Fahrenheit temperature ranges from 41° to 50°, inclusive, what is the range for Celsius temperature? Use interval notation to express this range.
115. If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if the number of outcomes that result in heads, satisfies Describe the number of outcomes that determine an unfair coin that is tossed 100 times.
In Exercises 116–127, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality.
116. You see a sign on a small moving truck that reads “Rent me for $20 a day.*” Oh, there’s an asterisk: There is an additional charge of $0.80 per mile. When you go online to reserve the truck, you are offered an unlimited mileage option for $60 a day. How many miles would you have to drive the truck in a day to make the unlimited mileage option the better deal?
117. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. For how many bridge crossings is toll-by-plate the better option?
118. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the electronic pass option is the better deal.
119. A city commission has proposed two tax bills. The first bill requires that a homeowner pay $1200 plus 0.5% of the assessed home value in taxes. The second bill requires taxes of $300 plus 0.9% of the assessed home value. What price range of home assessment would make the first bill a better deal?
120. A company designs and sells greeting cards. The weekly fixed cost is $10,000 and it costs $0.40 to create each card. The selling price is $2.00 per card. How many greeting cards must be designed and sold each week for the company to generate a profit?
121. A company manufactures and sells personalized stationery. The weekly fixed cost is $3000 and it costs $3.00 to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?
122. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
123. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
124. To earn an A in a course, you must have a final average of at least 90%. On the first four examinations, you have grades of 86%, 88%, 92%, and 84%. If the final examination counts as two grades, what must you get on the final to earn an A in the course?
125. On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90.
What must you get on the final to earn an A in the course?
By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than 80, you will lose your B in the course. Describe the grades on the final that will cause this to happen.
126. Parts for an automobile repair cost $254. The mechanic charges $65 per hour. If you receive an estimate for at least $351.50 and at most $481.50 for fixing the car, what is the time interval that the mechanic will be working on the job?
127. The toll to a bridge is $3.00. A three-month pass costs $7.50 and reduces the toll to $0.50. A six-month pass costs $30 and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?
Explaining the Concepts
128. When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?
129. Describe ways in which solving a linear inequality is similar to solving a linear equation.
130. Describe ways in which solving a linear inequality is different from solving a linear equation.
131. What is a compound inequality and how is it solved?
132. Describe how to solve an absolute value inequality involving the symbol Give an example.
133. Describe how to solve an absolute value inequality involving the symbol Give an example.
134. Explain why has no solution.
135. Describe the solution set of
Critical Thinking Exercises
Make Sense? In Exercises 136–139, determine whether each statement makes sense or does not make sense, and explain your reasoning.
136. I prefer interval notation over set-builder notation because it takes less space to write solution sets.
137. I can check inequalities by substituting 0 for the variable: When 0 belongs to the solution set, I should obtain a true statement, and when 0 does not belong to the solution set, I should obtain a false statement.
138. In an inequality such as I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.
139. I’ll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, is modeled by
In Exercises 140–143, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
140.
141.
142. The inequality is equivalent to
143. All irrational numbers satisfy
144. What’s wrong with this argument? Suppose x and represent two real numbers, where
The final inequality, , is impossible because we were initially given .
145. Write an absolute value inequality for which the interval shown is the solution.
146. Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without electronic passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.
Preview Exercises
Exercises 147–149 will help you prepare for the material covered in the first section of the next chapter.
147. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
148. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
149. If , find the value of y that corresponds to values of x for each integer starting with and ending with 2.
P.9: Exercise Set
P.9 Exercise Set
Practice Exercises
In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
In Exercises 15–26, use graphs to find each set.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In all exercises, use interval notation to express solution sets and graph each solution set on a number line.
In Exercises 27–48, solve each linear inequality.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–56, solve each compound inequality.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–92, solve each absolute value inequality.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–100, use interval notation to represent all values of x satisfying the given conditions.
93. and
94. and
95. and is at least 4.
96. and is at most 0.
97. and
98. and
99. and is at most 4.
100. and is at least 6.
101. When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.
102. When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.
Application Exercises
The graphs show that the three components of love, namely, passion, intimacy, and commitment, progress differently over time. Passion peaks early in a relationship and then declines. By contrast, intimacy and commitment build gradually. Use the graphs to solve Exercises 103–110.
Source: R. J. Sternberg. A Triangular Theory of Love, Psychological Review, 93, 119–135
103. Use interval notation to write an inequality that expresses for which years in a relationship intimacy is greater than commitment.
104. Use interval notation to write an inequality that expresses for which years in a relationship passion is greater than or equal to intimacy.
105. What is the relationship between passion and intimacy on the interval [5, 7)?
106. What is the relationship between intimacy and commitment on the interval [4, 7)?
107. What is the relationship between passion and commitment for
108. What is the relationship between passion and commitment for
109. What is the maximum level of intensity for passion? After how many years in a relationship does this occur?
110. After approximately how many years do levels of intensity for commitment exceed the maximum level of intensity for passion?
In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and 2012. Also shown is the percentage of households in which a person of faith is married to someone with no religion.
Source: General Social Survey, University of Chicago
models the percentage of U.S. households with an interfaith marriage, I, x years after 1988. The formula
models the percentage of U.S. households in which a person of faith is married to someone with no religion, N, x years after 1988.
Use these models to solve Exercises 111–112.
111.
In which years will more than 33% of U.S. households have an interfaith marriage?
In which years will more than 14% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage and more than 14% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage or more than 14% have a faith/no religion marriage?
112.
In which years will more than 34% of U.S. households have an interfaith marriage?
In which years will more than 15% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage and more than 15% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage or more than 15% have a faith/no religion marriage?
113. The formula for converting Fahrenheit temperature, to Celsius temperature, is
If Celsius temperature ranges from to , inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.
114. The formula for converting Celsius temperature, C, to Fahrenheit temperature, F, is
If Fahrenheit temperature ranges from 41° to 50°, inclusive, what is the range for Celsius temperature? Use interval notation to express this range.
115. If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if the number of outcomes that result in heads, satisfies Describe the number of outcomes that determine an unfair coin that is tossed 100 times.
In Exercises 116–127, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality.
116. You see a sign on a small moving truck that reads “Rent me for $20 a day.*” Oh, there’s an asterisk: There is an additional charge of $0.80 per mile. When you go online to reserve the truck, you are offered an unlimited mileage option for $60 a day. How many miles would you have to drive the truck in a day to make the unlimited mileage option the better deal?
117. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. For how many bridge crossings is toll-by-plate the better option?
118. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the electronic pass option is the better deal.
119. A city commission has proposed two tax bills. The first bill requires that a homeowner pay $1200 plus 0.5% of the assessed home value in taxes. The second bill requires taxes of $300 plus 0.9% of the assessed home value. What price range of home assessment would make the first bill a better deal?
120. A company designs and sells greeting cards. The weekly fixed cost is $10,000 and it costs $0.40 to create each card. The selling price is $2.00 per card. How many greeting cards must be designed and sold each week for the company to generate a profit?
121. A company manufactures and sells personalized stationery. The weekly fixed cost is $3000 and it costs $3.00 to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?
122. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
123. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
124. To earn an A in a course, you must have a final average of at least 90%. On the first four examinations, you have grades of 86%, 88%, 92%, and 84%. If the final examination counts as two grades, what must you get on the final to earn an A in the course?
125. On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90.
What must you get on the final to earn an A in the course?
By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than 80, you will lose your B in the course. Describe the grades on the final that will cause this to happen.
126. Parts for an automobile repair cost $254. The mechanic charges $65 per hour. If you receive an estimate for at least $351.50 and at most $481.50 for fixing the car, what is the time interval that the mechanic will be working on the job?
127. The toll to a bridge is $3.00. A three-month pass costs $7.50 and reduces the toll to $0.50. A six-month pass costs $30 and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?
Explaining the Concepts
128. When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?
129. Describe ways in which solving a linear inequality is similar to solving a linear equation.
130. Describe ways in which solving a linear inequality is different from solving a linear equation.
131. What is a compound inequality and how is it solved?
132. Describe how to solve an absolute value inequality involving the symbol Give an example.
133. Describe how to solve an absolute value inequality involving the symbol Give an example.
134. Explain why has no solution.
135. Describe the solution set of
Critical Thinking Exercises
Make Sense? In Exercises 136–139, determine whether each statement makes sense or does not make sense, and explain your reasoning.
136. I prefer interval notation over set-builder notation because it takes less space to write solution sets.
137. I can check inequalities by substituting 0 for the variable: When 0 belongs to the solution set, I should obtain a true statement, and when 0 does not belong to the solution set, I should obtain a false statement.
138. In an inequality such as I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.
139. I’ll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, is modeled by
In Exercises 140–143, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
140.
141.
142. The inequality is equivalent to
143. All irrational numbers satisfy
144. What’s wrong with this argument? Suppose x and represent two real numbers, where
The final inequality, , is impossible because we were initially given .
145. Write an absolute value inequality for which the interval shown is the solution.
146. Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without electronic passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.
Preview Exercises
Exercises 147–149 will help you prepare for the material covered in the first section of the next chapter.
147. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
148. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
149. If , find the value of y that corresponds to values of x for each integer starting with and ending with 2.
P.9: Exercise Set
P.9 Exercise Set
Practice Exercises
In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
In Exercises 15–26, use graphs to find each set.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In all exercises, use interval notation to express solution sets and graph each solution set on a number line.
In Exercises 27–48, solve each linear inequality.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–56, solve each compound inequality.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–92, solve each absolute value inequality.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–100, use interval notation to represent all values of x satisfying the given conditions.
93. and
94. and
95. and is at least 4.
96. and is at most 0.
97. and
98. and
99. and is at most 4.
100. and is at least 6.
101. When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.
102. When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.
Application Exercises
The graphs show that the three components of love, namely, passion, intimacy, and commitment, progress differently over time. Passion peaks early in a relationship and then declines. By contrast, intimacy and commitment build gradually. Use the graphs to solve Exercises 103–110.
Source: R. J. Sternberg. A Triangular Theory of Love, Psychological Review, 93, 119–135
103. Use interval notation to write an inequality that expresses for which years in a relationship intimacy is greater than commitment.
104. Use interval notation to write an inequality that expresses for which years in a relationship passion is greater than or equal to intimacy.
105. What is the relationship between passion and intimacy on the interval [5, 7)?
106. What is the relationship between intimacy and commitment on the interval [4, 7)?
107. What is the relationship between passion and commitment for
108. What is the relationship between passion and commitment for
109. What is the maximum level of intensity for passion? After how many years in a relationship does this occur?
110. After approximately how many years do levels of intensity for commitment exceed the maximum level of intensity for passion?
In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and 2012. Also shown is the percentage of households in which a person of faith is married to someone with no religion.
Source: General Social Survey, University of Chicago
models the percentage of U.S. households with an interfaith marriage, I, x years after 1988. The formula
models the percentage of U.S. households in which a person of faith is married to someone with no religion, N, x years after 1988.
Use these models to solve Exercises 111–112.
111.
In which years will more than 33% of U.S. households have an interfaith marriage?
In which years will more than 14% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage and more than 14% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 33% of households have an interfaith marriage or more than 14% have a faith/no religion marriage?
112.
In which years will more than 34% of U.S. households have an interfaith marriage?
In which years will more than 15% of U.S. households have a person of faith married to someone with no religion?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage and more than 15% have a faith/no religion marriage?
Based on your answers to parts (a) and (b), in which years will more than 34% of households have an interfaith marriage or more than 15% have a faith/no religion marriage?
113. The formula for converting Fahrenheit temperature, to Celsius temperature, is
If Celsius temperature ranges from to , inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.
114. The formula for converting Celsius temperature, C, to Fahrenheit temperature, F, is
If Fahrenheit temperature ranges from 41° to 50°, inclusive, what is the range for Celsius temperature? Use interval notation to express this range.
115. If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if the number of outcomes that result in heads, satisfies Describe the number of outcomes that determine an unfair coin that is tossed 100 times.
In Exercises 116–127, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality.
116. You see a sign on a small moving truck that reads “Rent me for $20 a day.*” Oh, there’s an asterisk: There is an additional charge of $0.80 per mile. When you go online to reserve the truck, you are offered an unlimited mileage option for $60 a day. How many miles would you have to drive the truck in a day to make the unlimited mileage option the better deal?
117. A transponder for a toll bridge costs $27.50. With the transponder, the toll is $5 each time you cross the bridge. The only other option is toll-by-plate, for which the toll is $6.25 each time you cross the bridge with an additional administrative fee of $1.25 for each crossing. For how many bridge crossings is toll-by-plate the better option?
118. An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the electronic pass option is the better deal.
119. A city commission has proposed two tax bills. The first bill requires that a homeowner pay $1200 plus 0.5% of the assessed home value in taxes. The second bill requires taxes of $300 plus 0.9% of the assessed home value. What price range of home assessment would make the first bill a better deal?
120. A company designs and sells greeting cards. The weekly fixed cost is $10,000 and it costs $0.40 to create each card. The selling price is $2.00 per card. How many greeting cards must be designed and sold each week for the company to generate a profit?
121. A company manufactures and sells personalized stationery. The weekly fixed cost is $3000 and it costs $3.00 to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?
122. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
123. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
124. To earn an A in a course, you must have a final average of at least 90%. On the first four examinations, you have grades of 86%, 88%, 92%, and 84%. If the final examination counts as two grades, what must you get on the final to earn an A in the course?
125. On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90.
What must you get on the final to earn an A in the course?
By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than 80, you will lose your B in the course. Describe the grades on the final that will cause this to happen.
126. Parts for an automobile repair cost $254. The mechanic charges $65 per hour. If you receive an estimate for at least $351.50 and at most $481.50 for fixing the car, what is the time interval that the mechanic will be working on the job?
127. The toll to a bridge is $3.00. A three-month pass costs $7.50 and reduces the toll to $0.50. A six-month pass costs $30 and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?
Explaining the Concepts
128. When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?
129. Describe ways in which solving a linear inequality is similar to solving a linear equation.
130. Describe ways in which solving a linear inequality is different from solving a linear equation.
131. What is a compound inequality and how is it solved?
132. Describe how to solve an absolute value inequality involving the symbol Give an example.
133. Describe how to solve an absolute value inequality involving the symbol Give an example.
134. Explain why has no solution.
135. Describe the solution set of
Critical Thinking Exercises
Make Sense? In Exercises 136–139, determine whether each statement makes sense or does not make sense, and explain your reasoning.
136. I prefer interval notation over set-builder notation because it takes less space to write solution sets.
137. I can check inequalities by substituting 0 for the variable: When 0 belongs to the solution set, I should obtain a true statement, and when 0 does not belong to the solution set, I should obtain a false statement.
138. In an inequality such as I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.
139. I’ll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, is modeled by
In Exercises 140–143, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
140.
141.
142. The inequality is equivalent to
143. All irrational numbers satisfy
144. What’s wrong with this argument? Suppose x and represent two real numbers, where
The final inequality, , is impossible because we were initially given .
145. Write an absolute value inequality for which the interval shown is the solution.
146. Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without electronic passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.
Preview Exercises
Exercises 147–149 will help you prepare for the material covered in the first section of the next chapter.
147. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
148. If , find the value of y that corresponds to values of x for each integer starting with and ending with 3.
149. If , find the value of y that corresponds to values of x for each integer starting with and ending with 2.
Chapter P: Review Exercises
Chapter P Review Exercises
You can use these review exercises, like the review exercises at the end of each chapter, to test your understanding of the chapter’s topics. However, you can also use these exercises as a prerequisite test to check your mastery of the fundamental algebra skills needed in this book.
P.1
In Exercises 1–2, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for and
3. You are riding along an expressway traveling x miles per hour. The formula
models the recommended safe distance, S, in feet, between your car and other cars on the expressway. What is the recommended safe distance when your speed is 60 miles per hour?
In Exercises 4–7, let and Find the indicated set.
4.
5.
6.
7.
8. Consider the set:
List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
In Exercises 9–11, rewrite each expression without absolute value bars.
9.
10.
11.
12. Express the distance between the numbers and 4 using absolute value. Then evaluate the absolute value.
In Exercises 13–18, state the name of the property illustrated.
13.
14.
15.
16.
17.
18.
In Exercises 19–22, simplify each algebraic expression.
19.
20.
21.
22.
23. The diversity index, from 0 (no diversity) to 100, measures the chance that two randomly selected people are a different race or ethnicity. The diversity index in the United States varies widely from region to region, from as high as 81 in Hawaii to as low as 11 in Vermont. The bar graph shows the national diversity index for the United States for five years in the period from 1980 through 2020.
The data in the graph can be modeled by the formula
where D is the national diversity index in the United States x years after 1980. According to the formula, what was the U.S. diversity index in 2010? How does this compare with the index displayed by the bar graph?
P.2
Evaluate each exponential expression in Exercises 24–27.
24.
25.
26.
27.
Simplify each exponential expression in Exercises 28–31.
28.
29.
30.
31.
In Exercises 32–33, write each number in decimal notation.
32.
33.
In Exercises 34–35, write each number in scientific notation.
34. 3,590,000
35. 0.00725
In Exercises 36–37, perform the indicated operation and write the answer in decimal notation.
36.
37.
38. The average salary of a professional baseball player is $4.1 million. (Source: Major League Baseball Player Association) Express this number in scientific notation.
39. The average salary of a nurse is $73,000. (Source: U.S. Department of Labor) Express this number in scientific notation.
40. Use your scientific notation answers from Exercises 38 and 39 to answer this question.
How many times greater is the average salary of a professional baseball player than the average salary of a nurse?
P.3
Use the product rule to simplify the expressions in Exercises 41–44. In Exercises 43–44, assume that variables represent nonnegative real numbers.
41.
42.
43.
44.
Use the quotient rule to simplify the expressions in Exercises 45–46.
45.
46.
In Exercises 47–49, add or subtract terms whenever possible.
47.
48.
49.
In Exercises 50–53, rationalize the denominator.
50.
51.
52.
53.
Evaluate each expression in Exercises 54–57 or indicate that the root is not a real number.
54.
55.
56.
57.
Simplify the radical expressions in Exercises 58–62.
58.
59.
60.
61.
62. (Assume that )
In Exercises 63–68, evaluate each expression.
63.
64.
65.
66.
67.
68.
In Exercises 69–71, simplify using properties of exponents.
69.
70.
71.
72. Simplify by reducing the index of the radical:
P.4
In Exercises 73–74, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
73.
74.
In Exercises 75–81, find each product.
75.
76.
77.
78.
79.
80.
81.
In Exercises 82–83, perform the indicated operations. Indicate the degree of the resulting polynomial.
82.
83.
In Exercises 84–88, find each product.
84.
85.
86.
87.
88.
P.5
In Exercises 89–105, factor completely, or state that the polynomial is prime.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
In Exercises 106–108, factor and simplify each algebraic expression.
106.
107.
108.
P.6
In Exercises 109–111, simplify each rational expression. Also, list all numbers that must be excluded from the domain.
109.
110.
111.
In Exercises 112–114, multiply or divide as indicated.
112.
113.
114.
In Exercises 115–120, add or subtract as indicated.
115.
116.
117.
118.
119.
120.
In Exercises 121–124, simplify each complex rational expression.
121.
122.
123.
124.
P.7
In Exercises 125–138, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
In Exercises 139–140, solve each formula for the specified variable.
139. for
140.
In Exercises 141–142, without solving the given quadratic equation, determine the number and type of solutions.
141.
142.
P.8
In Exercises 143–155, use the five-step strategy for solving word problems.
143. The Dog Ate My Calendar. The bar graph shows seven common excuses by college students for not meeting assignment deadlines. The bar heights represent the number of excuses for every 500 excuses that fall into each of these categories.
Source: Roig and Caso, “Lying and Cheating: Fraudulent Excuse Making, Cheating, and Plagiarism,” Journal of Psychology
For every 500 excuses, the number involving computer problems exceeds the number involving oversleeping by 10. The number involving illness exceeds the number involving oversleeping by 80. Combined, oversleeping, computer problems, and illness account for 270 excuses for not meeting assignment deadlines. For every 500 excuses, determine the number due to oversleeping, computer problems, and illness.
144. The bar graph shows the average price of a movie ticket for selected years from 1980 through 2019. The graph indicates that in 1980, the average movie ticket price was $2.69. For the period from 1980 through 2019, the price increased by approximately $0.17 per year. If this trend continues, by which year will the average price of a movie ticket be $10?
Sources: Motion Picture Association of America, National Association of Theater Owners (NATO), and Bureau of Labor Statistics (BLS)
145. You are choosing between two internet service providers. The first has a one-time installation and activation fee of $150 and a monthly charge of $60. The other offers the same services with a one-time fee of $30 and a monthly charge of $75. After how many months will the total costs for the two providers be the same?
146. An apartment complex has offered you a move-in special of 30% off the first month’s rent. If you pay $945 for the first month, what should you expect to pay for the second month when you must pay full price?
147. A real estate agent receives 3% commission on the sales price of a home. The agent has incurred $2125 in advertising and other expenses listing the home. If the agent would like to earn $9125 after expenses, what sales price is necessary?
148. You invested $9000 in two funds paying 1.7% and 1.9% annual interest. At the end of the year, the total interest from these investments was $166. How much was invested at each rate?
149. Last month you had a total of $5000 in interest-bearing balances on two credit cards. One card has a monthly interest rate of 1.75%, and the other has a monthly rate of 2.25%. If your total interest for the month was $94.75, what was the interest-bearing balance on each card?
150. The length of a rectangular field is 6 yards less than triple the width. If the perimeter of the field is 340 yards, what are its dimensions?
151. In 2015, there were 14,100 students at college A, with a projected enrollment increase of 1500 students per year. In the same year, there were 41,700 students at college B, with a projected enrollment decline of 800 students per year. In which year will the colleges have the same enrollment? What will be the enrollment in each college at that time?
152. An architect is allowed 15 square yards of floor space to add a small bedroom to a house. Because of the room’s design in relationship to the existing structure, the width of the rectangular floor must be 7 yards less than two times the length. Find the length and width of the rectangular floor that the architect is permitted.
153. A building casts a shadow that is double the length of its height. If the distance from the end of the shadow to the top of the building is 300 meters, how high is the building? Round to the nearest meter.
154. A painting measuring 10 inches by 16 inches is surrounded by a frame of uniform width. If the combined area of the painting and frame is 280 square inches, determine the width of the frame.
155. Club members equally share the cost of $1500 to charter a fishing boat. Shortly before the boat is to leave, four people decide not to go due to rough seas. As a result, the cost per person is increased by $100. How many people originally intended to go on the fishing trip?
P.9
In Exercises 156–158, express each interval in set-builder notation and graph the interval on a number line.
156.
157.
158.
In Exercises 159–162, use graphs to find each set.
159.
160.
161.
162.
In Exercises 163–172, solve each inequality. Other than inequalities with no solution, use interval notation to express solution sets and graph each solution set on a number line.
163.
164.
165.
166.
167.
168.
169.
170.
171.
172.
173. A car rental agency rents a certain car for $40 per day with unlimited mileage or $24 per day plus $0.20 per mile. How far can a customer drive this car per day for the $24 option to cost no more than the unlimited mileage option?
174. To receive a B in a course, you must have an average of at least 80% but less than 90% on five exams. Your grades on the first four exams were 95%, 79%, 91%, and 86%. What range of grades on the fifth exam will result in a B for the course?
Chapter P: Review Exercises
Chapter P Review Exercises
You can use these review exercises, like the review exercises at the end of each chapter, to test your understanding of the chapter’s topics. However, you can also use these exercises as a prerequisite test to check your mastery of the fundamental algebra skills needed in this book.
P.1
In Exercises 1–2, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for and
3. You are riding along an expressway traveling x miles per hour. The formula
models the recommended safe distance, S, in feet, between your car and other cars on the expressway. What is the recommended safe distance when your speed is 60 miles per hour?
In Exercises 4–7, let and Find the indicated set.
4.
5.
6.
7.
8. Consider the set:
List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
In Exercises 9–11, rewrite each expression without absolute value bars.
9.
10.
11.
12. Express the distance between the numbers and 4 using absolute value. Then evaluate the absolute value.
In Exercises 13–18, state the name of the property illustrated.
13.
14.
15.
16.
17.
18.
In Exercises 19–22, simplify each algebraic expression.
19.
20.
21.
22.
23. The diversity index, from 0 (no diversity) to 100, measures the chance that two randomly selected people are a different race or ethnicity. The diversity index in the United States varies widely from region to region, from as high as 81 in Hawaii to as low as 11 in Vermont. The bar graph shows the national diversity index for the United States for five years in the period from 1980 through 2020.
The data in the graph can be modeled by the formula
where D is the national diversity index in the United States x years after 1980. According to the formula, what was the U.S. diversity index in 2010? How does this compare with the index displayed by the bar graph?
P.2
Evaluate each exponential expression in Exercises 24–27.
24.
25.
26.
27.
Simplify each exponential expression in Exercises 28–31.
28.
29.
30.
31.
In Exercises 32–33, write each number in decimal notation.
32.
33.
In Exercises 34–35, write each number in scientific notation.
34. 3,590,000
35. 0.00725
In Exercises 36–37, perform the indicated operation and write the answer in decimal notation.
36.
37.
38. The average salary of a professional baseball player is $4.1 million. (Source: Major League Baseball Player Association) Express this number in scientific notation.
39. The average salary of a nurse is $73,000. (Source: U.S. Department of Labor) Express this number in scientific notation.
40. Use your scientific notation answers from Exercises 38 and 39 to answer this question.
How many times greater is the average salary of a professional baseball player than the average salary of a nurse?
P.3
Use the product rule to simplify the expressions in Exercises 41–44. In Exercises 43–44, assume that variables represent nonnegative real numbers.
41.
42.
43.
44.
Use the quotient rule to simplify the expressions in Exercises 45–46.
45.
46.
In Exercises 47–49, add or subtract terms whenever possible.
47.
48.
49.
In Exercises 50–53, rationalize the denominator.
50.
51.
52.
53.
Evaluate each expression in Exercises 54–57 or indicate that the root is not a real number.
54.
55.
56.
57.
Simplify the radical expressions in Exercises 58–62.
58.
59.
60.
61.
62. (Assume that )
In Exercises 63–68, evaluate each expression.
63.
64.
65.
66.
67.
68.
In Exercises 69–71, simplify using properties of exponents.
69.
70.
71.
72. Simplify by reducing the index of the radical:
P.4
In Exercises 73–74, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
73.
74.
In Exercises 75–81, find each product.
75.
76.
77.
78.
79.
80.
81.
In Exercises 82–83, perform the indicated operations. Indicate the degree of the resulting polynomial.
82.
83.
In Exercises 84–88, find each product.
84.
85.
86.
87.
88.
P.5
In Exercises 89–105, factor completely, or state that the polynomial is prime.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
In Exercises 106–108, factor and simplify each algebraic expression.
106.
107.
108.
P.6
In Exercises 109–111, simplify each rational expression. Also, list all numbers that must be excluded from the domain.
109.
110.
111.
In Exercises 112–114, multiply or divide as indicated.
112.
113.
114.
In Exercises 115–120, add or subtract as indicated.
115.
116.
117.
118.
119.
120.
In Exercises 121–124, simplify each complex rational expression.
121.
122.
123.
124.
P.7
In Exercises 125–138, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
In Exercises 139–140, solve each formula for the specified variable.
139. for
140.
In Exercises 141–142, without solving the given quadratic equation, determine the number and type of solutions.
141.
142.
P.8
In Exercises 143–155, use the five-step strategy for solving word problems.
143. The Dog Ate My Calendar. The bar graph shows seven common excuses by college students for not meeting assignment deadlines. The bar heights represent the number of excuses for every 500 excuses that fall into each of these categories.
Source: Roig and Caso, “Lying and Cheating: Fraudulent Excuse Making, Cheating, and Plagiarism,” Journal of Psychology
For every 500 excuses, the number involving computer problems exceeds the number involving oversleeping by 10. The number involving illness exceeds the number involving oversleeping by 80. Combined, oversleeping, computer problems, and illness account for 270 excuses for not meeting assignment deadlines. For every 500 excuses, determine the number due to oversleeping, computer problems, and illness.
144. The bar graph shows the average price of a movie ticket for selected years from 1980 through 2019. The graph indicates that in 1980, the average movie ticket price was $2.69. For the period from 1980 through 2019, the price increased by approximately $0.17 per year. If this trend continues, by which year will the average price of a movie ticket be $10?
Sources: Motion Picture Association of America, National Association of Theater Owners (NATO), and Bureau of Labor Statistics (BLS)
145. You are choosing between two internet service providers. The first has a one-time installation and activation fee of $150 and a monthly charge of $60. The other offers the same services with a one-time fee of $30 and a monthly charge of $75. After how many months will the total costs for the two providers be the same?
146. An apartment complex has offered you a move-in special of 30% off the first month’s rent. If you pay $945 for the first month, what should you expect to pay for the second month when you must pay full price?
147. A real estate agent receives 3% commission on the sales price of a home. The agent has incurred $2125 in advertising and other expenses listing the home. If the agent would like to earn $9125 after expenses, what sales price is necessary?
148. You invested $9000 in two funds paying 1.7% and 1.9% annual interest. At the end of the year, the total interest from these investments was $166. How much was invested at each rate?
149. Last month you had a total of $5000 in interest-bearing balances on two credit cards. One card has a monthly interest rate of 1.75%, and the other has a monthly rate of 2.25%. If your total interest for the month was $94.75, what was the interest-bearing balance on each card?
150. The length of a rectangular field is 6 yards less than triple the width. If the perimeter of the field is 340 yards, what are its dimensions?
151. In 2015, there were 14,100 students at college A, with a projected enrollment increase of 1500 students per year. In the same year, there were 41,700 students at college B, with a projected enrollment decline of 800 students per year. In which year will the colleges have the same enrollment? What will be the enrollment in each college at that time?
152. An architect is allowed 15 square yards of floor space to add a small bedroom to a house. Because of the room’s design in relationship to the existing structure, the width of the rectangular floor must be 7 yards less than two times the length. Find the length and width of the rectangular floor that the architect is permitted.
153. A building casts a shadow that is double the length of its height. If the distance from the end of the shadow to the top of the building is 300 meters, how high is the building? Round to the nearest meter.
154. A painting measuring 10 inches by 16 inches is surrounded by a frame of uniform width. If the combined area of the painting and frame is 280 square inches, determine the width of the frame.
155. Club members equally share the cost of $1500 to charter a fishing boat. Shortly before the boat is to leave, four people decide not to go due to rough seas. As a result, the cost per person is increased by $100. How many people originally intended to go on the fishing trip?
P.9
In Exercises 156–158, express each interval in set-builder notation and graph the interval on a number line.
156.
157.
158.
In Exercises 159–162, use graphs to find each set.
159.
160.
161.
162.
In Exercises 163–172, solve each inequality. Other than inequalities with no solution, use interval notation to express solution sets and graph each solution set on a number line.
163.
164.
165.
166.
167.
168.
169.
170.
171.
172.
173. A car rental agency rents a certain car for $40 per day with unlimited mileage or $24 per day plus $0.20 per mile. How far can a customer drive this car per day for the $24 option to cost no more than the unlimited mileage option?
174. To receive a B in a course, you must have an average of at least 80% but less than 90% on five exams. Your grades on the first four exams were 95%, 79%, 91%, and 86%. What range of grades on the fifth exam will result in a B for the course?
Chapter P: Review Exercises
Chapter P Review Exercises
You can use these review exercises, like the review exercises at the end of each chapter, to test your understanding of the chapter’s topics. However, you can also use these exercises as a prerequisite test to check your mastery of the fundamental algebra skills needed in this book.
P.1
In Exercises 1–2, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for and
3. You are riding along an expressway traveling x miles per hour. The formula
models the recommended safe distance, S, in feet, between your car and other cars on the expressway. What is the recommended safe distance when your speed is 60 miles per hour?
In Exercises 4–7, let and Find the indicated set.
4.
5.
6.
7.
8. Consider the set:
List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
In Exercises 9–11, rewrite each expression without absolute value bars.
9.
10.
11.
12. Express the distance between the numbers and 4 using absolute value. Then evaluate the absolute value.
In Exercises 13–18, state the name of the property illustrated.
13.
14.
15.
16.
17.
18.
In Exercises 19–22, simplify each algebraic expression.
19.
20.
21.
22.
23. The diversity index, from 0 (no diversity) to 100, measures the chance that two randomly selected people are a different race or ethnicity. The diversity index in the United States varies widely from region to region, from as high as 81 in Hawaii to as low as 11 in Vermont. The bar graph shows the national diversity index for the United States for five years in the period from 1980 through 2020.
The data in the graph can be modeled by the formula
where D is the national diversity index in the United States x years after 1980. According to the formula, what was the U.S. diversity index in 2010? How does this compare with the index displayed by the bar graph?
P.2
Evaluate each exponential expression in Exercises 24–27.
24.
25.
26.
27.
Simplify each exponential expression in Exercises 28–31.
28.
29.
30.
31.
In Exercises 32–33, write each number in decimal notation.
32.
33.
In Exercises 34–35, write each number in scientific notation.
34. 3,590,000
35. 0.00725
In Exercises 36–37, perform the indicated operation and write the answer in decimal notation.
36.
37.
38. The average salary of a professional baseball player is $4.1 million. (Source: Major League Baseball Player Association) Express this number in scientific notation.
39. The average salary of a nurse is $73,000. (Source: U.S. Department of Labor) Express this number in scientific notation.
40. Use your scientific notation answers from Exercises 38 and 39 to answer this question.
How many times greater is the average salary of a professional baseball player than the average salary of a nurse?
P.3
Use the product rule to simplify the expressions in Exercises 41–44. In Exercises 43–44, assume that variables represent nonnegative real numbers.
41.
42.
43.
44.
Use the quotient rule to simplify the expressions in Exercises 45–46.
45.
46.
In Exercises 47–49, add or subtract terms whenever possible.
47.
48.
49.
In Exercises 50–53, rationalize the denominator.
50.
51.
52.
53.
Evaluate each expression in Exercises 54–57 or indicate that the root is not a real number.
54.
55.
56.
57.
Simplify the radical expressions in Exercises 58–62.
58.
59.
60.
61.
62. (Assume that )
In Exercises 63–68, evaluate each expression.
63.
64.
65.
66.
67.
68.
In Exercises 69–71, simplify using properties of exponents.
69.
70.
71.
72. Simplify by reducing the index of the radical:
P.4
In Exercises 73–74, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
73.
74.
In Exercises 75–81, find each product.
75.
76.
77.
78.
79.
80.
81.
In Exercises 82–83, perform the indicated operations. Indicate the degree of the resulting polynomial.
82.
83.
In Exercises 84–88, find each product.
84.
85.
86.
87.
88.
P.5
In Exercises 89–105, factor completely, or state that the polynomial is prime.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
In Exercises 106–108, factor and simplify each algebraic expression.
106.
107.
108.
P.6
In Exercises 109–111, simplify each rational expression. Also, list all numbers that must be excluded from the domain.
109.
110.
111.
In Exercises 112–114, multiply or divide as indicated.
112.
113.
114.
In Exercises 115–120, add or subtract as indicated.
115.
116.
117.
118.
119.
120.
In Exercises 121–124, simplify each complex rational expression.
121.
122.
123.
124.
P.7
In Exercises 125–138, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
In Exercises 139–140, solve each formula for the specified variable.
139. for
140.
In Exercises 141–142, without solving the given quadratic equation, determine the number and type of solutions.
141.
142.
P.8
In Exercises 143–155, use the five-step strategy for solving word problems.
143. The Dog Ate My Calendar. The bar graph shows seven common excuses by college students for not meeting assignment deadlines. The bar heights represent the number of excuses for every 500 excuses that fall into each of these categories.
Source: Roig and Caso, “Lying and Cheating: Fraudulent Excuse Making, Cheating, and Plagiarism,” Journal of Psychology
For every 500 excuses, the number involving computer problems exceeds the number involving oversleeping by 10. The number involving illness exceeds the number involving oversleeping by 80. Combined, oversleeping, computer problems, and illness account for 270 excuses for not meeting assignment deadlines. For every 500 excuses, determine the number due to oversleeping, computer problems, and illness.
144. The bar graph shows the average price of a movie ticket for selected years from 1980 through 2019. The graph indicates that in 1980, the average movie ticket price was $2.69. For the period from 1980 through 2019, the price increased by approximately $0.17 per year. If this trend continues, by which year will the average price of a movie ticket be $10?
Sources: Motion Picture Association of America, National Association of Theater Owners (NATO), and Bureau of Labor Statistics (BLS)
145. You are choosing between two internet service providers. The first has a one-time installation and activation fee of $150 and a monthly charge of $60. The other offers the same services with a one-time fee of $30 and a monthly charge of $75. After how many months will the total costs for the two providers be the same?
146. An apartment complex has offered you a move-in special of 30% off the first month’s rent. If you pay $945 for the first month, what should you expect to pay for the second month when you must pay full price?
147. A real estate agent receives 3% commission on the sales price of a home. The agent has incurred $2125 in advertising and other expenses listing the home. If the agent would like to earn $9125 after expenses, what sales price is necessary?
148. You invested $9000 in two funds paying 1.7% and 1.9% annual interest. At the end of the year, the total interest from these investments was $166. How much was invested at each rate?
149. Last month you had a total of $5000 in interest-bearing balances on two credit cards. One card has a monthly interest rate of 1.75%, and the other has a monthly rate of 2.25%. If your total interest for the month was $94.75, what was the interest-bearing balance on each card?
150. The length of a rectangular field is 6 yards less than triple the width. If the perimeter of the field is 340 yards, what are its dimensions?
151. In 2015, there were 14,100 students at college A, with a projected enrollment increase of 1500 students per year. In the same year, there were 41,700 students at college B, with a projected enrollment decline of 800 students per year. In which year will the colleges have the same enrollment? What will be the enrollment in each college at that time?
152. An architect is allowed 15 square yards of floor space to add a small bedroom to a house. Because of the room’s design in relationship to the existing structure, the width of the rectangular floor must be 7 yards less than two times the length. Find the length and width of the rectangular floor that the architect is permitted.
153. A building casts a shadow that is double the length of its height. If the distance from the end of the shadow to the top of the building is 300 meters, how high is the building? Round to the nearest meter.
154. A painting measuring 10 inches by 16 inches is surrounded by a frame of uniform width. If the combined area of the painting and frame is 280 square inches, determine the width of the frame.
155. Club members equally share the cost of $1500 to charter a fishing boat. Shortly before the boat is to leave, four people decide not to go due to rough seas. As a result, the cost per person is increased by $100. How many people originally intended to go on the fishing trip?
P.9
In Exercises 156–158, express each interval in set-builder notation and graph the interval on a number line.
156.
157.
158.
In Exercises 159–162, use graphs to find each set.
159.
160.
161.
162.
In Exercises 163–172, solve each inequality. Other than inequalities with no solution, use interval notation to express solution sets and graph each solution set on a number line.
163.
164.
165.
166.
167.
168.
169.
170.
171.
172.
173. A car rental agency rents a certain car for $40 per day with unlimited mileage or $24 per day plus $0.20 per mile. How far can a customer drive this car per day for the $24 option to cost no more than the unlimited mileage option?
174. To receive a B in a course, you must have an average of at least 80% but less than 90% on five exams. Your grades on the first four exams were 95%, 79%, 91%, and 86%. What range of grades on the fifth exam will result in a B for the course?
Chapter P: Review Exercises
Chapter P Review Exercises
You can use these review exercises, like the review exercises at the end of each chapter, to test your understanding of the chapter’s topics. However, you can also use these exercises as a prerequisite test to check your mastery of the fundamental algebra skills needed in this book.
P.1
In Exercises 1–2, evaluate each algebraic expression for the given value or values of the variable(s).
1. for
2. for and
3. You are riding along an expressway traveling x miles per hour. The formula
models the recommended safe distance, S, in feet, between your car and other cars on the expressway. What is the recommended safe distance when your speed is 60 miles per hour?
In Exercises 4–7, let and Find the indicated set.
4.
5.
6.
7.
8. Consider the set:
List all numbers from the set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers.
In Exercises 9–11, rewrite each expression without absolute value bars.
9.
10.
11.
12. Express the distance between the numbers and 4 using absolute value. Then evaluate the absolute value.
In Exercises 13–18, state the name of the property illustrated.
13.
14.
15.
16.
17.
18.
In Exercises 19–22, simplify each algebraic expression.
19.
20.
21.
22.
23. The diversity index, from 0 (no diversity) to 100, measures the chance that two randomly selected people are a different race or ethnicity. The diversity index in the United States varies widely from region to region, from as high as 81 in Hawaii to as low as 11 in Vermont. The bar graph shows the national diversity index for the United States for five years in the period from 1980 through 2020.
The data in the graph can be modeled by the formula
where D is the national diversity index in the United States x years after 1980. According to the formula, what was the U.S. diversity index in 2010? How does this compare with the index displayed by the bar graph?
P.2
Evaluate each exponential expression in Exercises 24–27.
24.
25.
26.
27.
Simplify each exponential expression in Exercises 28–31.
28.
29.
30.
31.
In Exercises 32–33, write each number in decimal notation.
32.
33.
In Exercises 34–35, write each number in scientific notation.
34. 3,590,000
35. 0.00725
In Exercises 36–37, perform the indicated operation and write the answer in decimal notation.
36.
37.
38. The average salary of a professional baseball player is $4.1 million. (Source: Major League Baseball Player Association) Express this number in scientific notation.
39. The average salary of a nurse is $73,000. (Source: U.S. Department of Labor) Express this number in scientific notation.
40. Use your scientific notation answers from Exercises 38 and 39 to answer this question.
How many times greater is the average salary of a professional baseball player than the average salary of a nurse?
P.3
Use the product rule to simplify the expressions in Exercises 41–44. In Exercises 43–44, assume that variables represent nonnegative real numbers.
41.
42.
43.
44.
Use the quotient rule to simplify the expressions in Exercises 45–46.
45.
46.
In Exercises 47–49, add or subtract terms whenever possible.
47.
48.
49.
In Exercises 50–53, rationalize the denominator.
50.
51.
52.
53.
Evaluate each expression in Exercises 54–57 or indicate that the root is not a real number.
54.
55.
56.
57.
Simplify the radical expressions in Exercises 58–62.
58.
59.
60.
61.
62. (Assume that )
In Exercises 63–68, evaluate each expression.
63.
64.
65.
66.
67.
68.
In Exercises 69–71, simplify using properties of exponents.
69.
70.
71.
72. Simplify by reducing the index of the radical:
P.4
In Exercises 73–74, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree.
73.
74.
In Exercises 75–81, find each product.
75.
76.
77.
78.
79.
80.
81.
In Exercises 82–83, perform the indicated operations. Indicate the degree of the resulting polynomial.
82.
83.
In Exercises 84–88, find each product.
84.
85.
86.
87.
88.
P.5
In Exercises 89–105, factor completely, or state that the polynomial is prime.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
In Exercises 106–108, factor and simplify each algebraic expression.
106.
107.
108.
P.6
In Exercises 109–111, simplify each rational expression. Also, list all numbers that must be excluded from the domain.
109.
110.
111.
In Exercises 112–114, multiply or divide as indicated.
112.
113.
114.
In Exercises 115–120, add or subtract as indicated.
115.
116.
117.
118.
119.
120.
In Exercises 121–124, simplify each complex rational expression.
121.
122.
123.
124.
P.7
In Exercises 125–138, solve each equation.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
In Exercises 139–140, solve each formula for the specified variable.
139. for
140.
In Exercises 141–142, without solving the given quadratic equation, determine the number and type of solutions.
141.
142.
P.8
In Exercises 143–155, use the five-step strategy for solving word problems.
143. The Dog Ate My Calendar. The bar graph shows seven common excuses by college students for not meeting assignment deadlines. The bar heights represent the number of excuses for every 500 excuses that fall into each of these categories.
Source: Roig and Caso, “Lying and Cheating: Fraudulent Excuse Making, Cheating, and Plagiarism,” Journal of Psychology
For every 500 excuses, the number involving computer problems exceeds the number involving oversleeping by 10. The number involving illness exceeds the number involving oversleeping by 80. Combined, oversleeping, computer problems, and illness account for 270 excuses for not meeting assignment deadlines. For every 500 excuses, determine the number due to oversleeping, computer problems, and illness.
144. The bar graph shows the average price of a movie ticket for selected years from 1980 through 2019. The graph indicates that in 1980, the average movie ticket price was $2.69. For the period from 1980 through 2019, the price increased by approximately $0.17 per year. If this trend continues, by which year will the average price of a movie ticket be $10?
Sources: Motion Picture Association of America, National Association of Theater Owners (NATO), and Bureau of Labor Statistics (BLS)
145. You are choosing between two internet service providers. The first has a one-time installation and activation fee of $150 and a monthly charge of $60. The other offers the same services with a one-time fee of $30 and a monthly charge of $75. After how many months will the total costs for the two providers be the same?
146. An apartment complex has offered you a move-in special of 30% off the first month’s rent. If you pay $945 for the first month, what should you expect to pay for the second month when you must pay full price?
147. A real estate agent receives 3% commission on the sales price of a home. The agent has incurred $2125 in advertising and other expenses listing the home. If the agent would like to earn $9125 after expenses, what sales price is necessary?
148. You invested $9000 in two funds paying 1.7% and 1.9% annual interest. At the end of the year, the total interest from these investments was $166. How much was invested at each rate?
149. Last month you had a total of $5000 in interest-bearing balances on two credit cards. One card has a monthly interest rate of 1.75%, and the other has a monthly rate of 2.25%. If your total interest for the month was $94.75, what was the interest-bearing balance on each card?
150. The length of a rectangular field is 6 yards less than triple the width. If the perimeter of the field is 340 yards, what are its dimensions?
151. In 2015, there were 14,100 students at college A, with a projected enrollment increase of 1500 students per year. In the same year, there were 41,700 students at college B, with a projected enrollment decline of 800 students per year. In which year will the colleges have the same enrollment? What will be the enrollment in each college at that time?
152. An architect is allowed 15 square yards of floor space to add a small bedroom to a house. Because of the room’s design in relationship to the existing structure, the width of the rectangular floor must be 7 yards less than two times the length. Find the length and width of the rectangular floor that the architect is permitted.
153. A building casts a shadow that is double the length of its height. If the distance from the end of the shadow to the top of the building is 300 meters, how high is the building? Round to the nearest meter.
154. A painting measuring 10 inches by 16 inches is surrounded by a frame of uniform width. If the combined area of the painting and frame is 280 square inches, determine the width of the frame.
155. Club members equally share the cost of $1500 to charter a fishing boat. Shortly before the boat is to leave, four people decide not to go due to rough seas. As a result, the cost per person is increased by $100. How many people originally intended to go on the fishing trip?
P.9
In Exercises 156–158, express each interval in set-builder notation and graph the interval on a number line.
156.
157.
158.
In Exercises 159–162, use graphs to find each set.
159.
160.
161.
162.
In Exercises 163–172, solve each inequality. Other than inequalities with no solution, use interval notation to express solution sets and graph each solution set on a number line.
163.
164.
165.
166.
167.
168.
169.
170.
171.
172.
173. A car rental agency rents a certain car for $40 per day with unlimited mileage or $24 per day plus $0.20 per mile. How far can a customer drive this car per day for the $24 option to cost no more than the unlimited mileage option?
174. To receive a B in a course, you must have an average of at least 80% but less than 90% on five exams. Your grades on the first four exams were 95%, 79%, 91%, and 86%. What range of grades on the fifth exam will result in a B for the course?
Chapter P: Test
Chapter P Test
You can check your answers against those at the back of the book. Step-by-step solutions are found in the Chapter Test Prep Videos available in MyLab Math and at youtube.com/user/pearsonmathstats (playlist “Blitzer Precalculus 7e”).
In Exercises 1–18, simplify the given expression or perform the indicated operation (and simplify, if possible), whichever is appropriate.
1.
2.
3.
4.
5.
6. (Assume that
7.
8.
9.
10.
11. (Express the answer in scientific notation.)
12.
13.
14.
15.
16.
17.
18.
In Exercises 19–24, factor completely, or state that the polynomial is prime.
19.
20.
21.
22.
23.
24.
25. Factor and simplify:
26. List all the rational numbers in this set:
In Exercises 27–28, state the name of the property illustrated.
27.
28.
29. Express in scientific notation: 0.00076.
30. Evaluate:
31. The human body contains approximately microliters of blood for every pound of body weight. Each microliter of blood contains approximately red blood cells. Express in scientific notation the approximate number of red blood cells in the body of a 180-pound person.
32. Big (Lack of) Men on Campus In 2007, 135 women received bachelor’s degrees for every 100 men. According to the U.S. Department of Education, that gender imbalance has widened, as shown by the bar graph.
According to the first formula, what percentage of bachelor’s degrees were awarded to men in 2003? Does this underestimate or overestimate the actual percent shown by the bar graph? By how much?
Use the given formulas to write a new formula with a rational expression that models the ratio of the percentage of bachelor’s degrees received by men to the percentage received by women n years after 1989. Name this new mathematical model R, for ratio.
Use the formula for R to find the ratio of bachelor’s degrees received by men to degrees received by women in 2017. According to the model, how many women received bachelor’s degrees for every two men in 2014? How well does this describe the data shown by the graph?
In Exercises 33–47, solve each equation or inequality. Use interval notation to express solution sets of inequalities and graph these solution sets on a number line.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
In Exercises 48–50, solve each formula for the specified variable.
48. for h
49. for x
50. for a
The graphs show the amount being paid in Social Security benefits and the amount going into the system. All data are expressed in billions of dollars. Amounts from 2016 through 2024 are projections.
Exercises 51–53 are based on the data shown by the graphs.
51. In 2004, the system’s income was $575 billion, projected to increase at an average rate of $43 billion per year. In which year was the system’s income $1177 billion?
52. The data for the system’s outflow can be modeled by the formula
where represents the amount paid in benefits, in billions of dollars, x years after 2004. According to this model, when was the amount paid in benefits $1177 billion? Round to the nearest year.
53. How well do your answers to Exercises 51 and 52 model the data shown by the graphs?
54. Here’s Looking at You. According to University of Texas economist Daniel Hamermesh (Beauty Pays: Why Attractive People Are More Successful), strikingly attractive and good-looking men and women can expect to earn an average of $230,000 more in a lifetime than a person who is homely or plain. The bar graph shows the distribution of looks for American men and women, ranging from homely to strikingly attractive.
The percentage of average-looking men exceeds the percentage of strikingly attractive men by 57. The percentage of good-looking men exceeds the percentage of strikingly attractive men by 25. A total of 88% of American men range between average-looking, good-looking, and strikingly attractive. Find the percentage of men who fall within each of these three categories of looks.
55. The costs for two different kinds of heating systems for a small home are given in the following table. After how many years will total costs for solar heating and electric heating be the same? What will be the cost at that time?
System
Cost to Install
Operating Cost/Year
Solar
$29,700
$150
Electric
$5000
$1100
56. You invested $10,000 in two accounts paying 1.3% and 1.7% annual interest. At the end of the year, the total interest from these investments was $158. How much was invested at each rate?
57. The length of a rectangular carpet is 4 feet greater than twice its width. If the area is 48 square feet, find the carpet’s length and width.
58. A vertical pole is to be supported by a wire that is 26 feet long and anchored 24 feet from the base of the pole. How far up the pole should the wire be attached?
59. After a 60% reduction, a jacket sold for $52. What was the jacket’s price before the reduction?
60. A group of people would like to buy a vacation cabin for $600,000, sharing the cost equally. If they could find five more people to join them, each person’s share would be reduced by $6000. How many people are in the group?
61. You are choosing between two gyms. The first has a one-time membership fee of $30 and charges $10 per month. The second has a one-time fee of $10 and charges $15 per month. How many months of membership make the second gym the better deal?
Chapter P: Test
Chapter P Test
You can check your answers against those at the back of the book. Step-by-step solutions are found in the Chapter Test Prep Videos available in MyLab Math and at youtube.com/user/pearsonmathstats (playlist “Blitzer Precalculus 7e”).
In Exercises 1–18, simplify the given expression or perform the indicated operation (and simplify, if possible), whichever is appropriate.
1.
2.
3.
4.
5.
6. (Assume that
7.
8.
9.
10.
11. (Express the answer in scientific notation.)
12.
13.
14.
15.
16.
17.
18.
In Exercises 19–24, factor completely, or state that the polynomial is prime.
19.
20.
21.
22.
23.
24.
25. Factor and simplify:
26. List all the rational numbers in this set:
In Exercises 27–28, state the name of the property illustrated.
27.
28.
29. Express in scientific notation: 0.00076.
30. Evaluate:
31. The human body contains approximately microliters of blood for every pound of body weight. Each microliter of blood contains approximately red blood cells. Express in scientific notation the approximate number of red blood cells in the body of a 180-pound person.
32. Big (Lack of) Men on Campus In 2007, 135 women received bachelor’s degrees for every 100 men. According to the U.S. Department of Education, that gender imbalance has widened, as shown by the bar graph.
According to the first formula, what percentage of bachelor’s degrees were awarded to men in 2003? Does this underestimate or overestimate the actual percent shown by the bar graph? By how much?
Use the given formulas to write a new formula with a rational expression that models the ratio of the percentage of bachelor’s degrees received by men to the percentage received by women n years after 1989. Name this new mathematical model R, for ratio.
Use the formula for R to find the ratio of bachelor’s degrees received by men to degrees received by women in 2017. According to the model, how many women received bachelor’s degrees for every two men in 2014? How well does this describe the data shown by the graph?
In Exercises 33–47, solve each equation or inequality. Use interval notation to express solution sets of inequalities and graph these solution sets on a number line.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
In Exercises 48–50, solve each formula for the specified variable.
48. for h
49. for x
50. for a
The graphs show the amount being paid in Social Security benefits and the amount going into the system. All data are expressed in billions of dollars. Amounts from 2016 through 2024 are projections.
Exercises 51–53 are based on the data shown by the graphs.
51. In 2004, the system’s income was $575 billion, projected to increase at an average rate of $43 billion per year. In which year was the system’s income $1177 billion?
52. The data for the system’s outflow can be modeled by the formula
where represents the amount paid in benefits, in billions of dollars, x years after 2004. According to this model, when was the amount paid in benefits $1177 billion? Round to the nearest year.
53. How well do your answers to Exercises 51 and 52 model the data shown by the graphs?
54. Here’s Looking at You. According to University of Texas economist Daniel Hamermesh (Beauty Pays: Why Attractive People Are More Successful), strikingly attractive and good-looking men and women can expect to earn an average of $230,000 more in a lifetime than a person who is homely or plain. The bar graph shows the distribution of looks for American men and women, ranging from homely to strikingly attractive.
The percentage of average-looking men exceeds the percentage of strikingly attractive men by 57. The percentage of good-looking men exceeds the percentage of strikingly attractive men by 25. A total of 88% of American men range between average-looking, good-looking, and strikingly attractive. Find the percentage of men who fall within each of these three categories of looks.
55. The costs for two different kinds of heating systems for a small home are given in the following table. After how many years will total costs for solar heating and electric heating be the same? What will be the cost at that time?
System
Cost to Install
Operating Cost/Year
Solar
$29,700
$150
Electric
$5000
$1100
56. You invested $10,000 in two accounts paying 1.3% and 1.7% annual interest. At the end of the year, the total interest from these investments was $158. How much was invested at each rate?
57. The length of a rectangular carpet is 4 feet greater than twice its width. If the area is 48 square feet, find the carpet’s length and width.
58. A vertical pole is to be supported by a wire that is 26 feet long and anchored 24 feet from the base of the pole. How far up the pole should the wire be attached?
59. After a 60% reduction, a jacket sold for $52. What was the jacket’s price before the reduction?
60. A group of people would like to buy a vacation cabin for $600,000, sharing the cost equally. If they could find five more people to join them, each person’s share would be reduced by $6000. How many people are in the group?
61. You are choosing between two gyms. The first has a one-time membership fee of $30 and charges $10 per month. The second has a one-time fee of $10 and charges $15 per month. How many months of membership make the second gym the better deal?
Chapter 1: Functions and Graphs
Chapter1 Functions and Graphs
A vast expanse of open water at the top of our world was once covered with ice. The melting of the Arctic ice caps has forced polar bears to swim as far as 40 miles, causing them to drown in significant numbers. Such deaths were rare in the past.
There is strong scientific consensus that human activities are changing the Earth’s climate. Scientists now believe that there is a striking correlation between atmospheric carbon dioxide concentration and global temperature. As both of these variables increase at significant rates, there are warnings of a planetary emergency that threatens to condemn coming generations to a catastrophically diminished future.*
In this chapter, you’ll learn to approach our climate crisis mathematically by creating formulas, called functions, that model data for average global temperature and carbon dioxide concentration over time. Understanding the concept of a function will give you a new perspective on many situations, ranging from climate change to using mathematics in a way that is similar to making a movie.
*Sources: Al Gore, An Inconvenient Truth, Rodale, 2006; Time, April 3, 2006; Rolling Stone, September 26, 2013
Here’s where you’ll find these applications:
A mathematical model involving global warming is developed in Example 9 in Section 1.4.
Using mathematics in a way that is similar to making a movie is discussed in the Blitzer Bonus here.
The beginning of the seventeenth century was a time of innovative ideas and enormous intellectual progress in Europe. English theatergoers enjoyed a succession of exciting new plays by Shakespeare. William Harvey proposed the radical notion that the heart was a pump for blood rather than the center of emotion. Galileo, with his newfangled invention called the telescope, supported the theory of Polish astronomer Copernicus that the Sun, not the Earth, was the center of the solar system. Monteverdi was writing the world’s first grand operas. French mathematicians Pascal and Fermat invented a new field of mathematics called probability theory.
Into this arena of intellectual electricity stepped French aristocrat René Descartes (1596–1650). Descartes (pronounced “day cart”), propelled by the creativity surrounding him, developed a new branch of mathematics that brought together algebra and geometry in a unified way—a way that visualized numbers as points on a graph, equations as geometric figures, and geometric figures as equations. This new branch of mathematics, called analytic geometry, established Descartes as one of the founders of modern thought and among the most original mathematicians and philosophers of any age. We begin this section by looking at Descartes’s deceptively simple idea, called the rectangular coordinate system or (in his honor) the Cartesian coordinate system.
Objective 1: Plot points in the rectangular coordinate system
Points and Ordered Pairs
Objective 1 Plot points in the rectangular coordinate system.
Watch Video
Descartes used two number lines that intersect at right angles at their zero points, as shown in Figure 1.1. The horizontal number line is the x-axis. The vertical number line is the y-axis. The point of intersection of these axes is their zero points, called the origin. Positive numbers are shown to the right and above the origin. Negative numbers are shown to the left and below the origin. The axes divide the plane into four quarters, called quadrants. The points located on the axes are not in any quadrant.
Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers, . Examples of such pairs are and . The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number in each pair, called the y-coordinate, denotes vertical distance and direction along a line parallel to the y-axis or along the y-axis itself.
Figure 1.2 shows how we plot, or locate, the points corresponding to the ordered pairs and . We plot by going 5 units from 0 to the left along the x-axis. Then we go 3 units up parallel to the y-axis. We plot by going 3 units from 0 to the right along the x-axis and 5 units down parallel to the y-axis. The phrase “the points corresponding to the ordered pairs and ” is often abbreviated as “the points and .”
Objective 2: Graph equations in the rectangular coordinate system
Graphs of Equations
Objective 2 Graph equations in the rectangular coordinate system.
Watch Video
A relationship between two quantities can be expressed as an equation in two variables, such as
A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement. For example, consider the equation and the ordered pair . When 3 is substituted for x and is substituted for y, we obtain the statement , or , or . Because this statement is true, the ordered pair is a solution of the equation . We also say that satisfies the equation.
We can generate as many ordered-pair solutions as desired to by substituting numbers for x and then finding the corresponding values for y. For example, suppose we let
The graph of an equation in two variables is the set of all points whose coordinates satisfy the equation. One method for graphing such equations is the point-plotting method. First, we find several ordered pairs that are solutions of the equation. Next, we plot these ordered pairs as points in the rectangular coordinate system. Finally, we connect the points with a smooth curve or line. This often gives us a picture of all ordered pairs that satisfy the equation.
Example 2 Graphing an Equation Using the Point-Plotting Method
Graph . Select integers for x, starting with and ending with 3.
Solution
For each value of x, we find the corresponding value for y.
Now we plot the seven points and join them with a smooth curve, as shown in Figure 1.4. The graph of is a curve where the part of the graph to the right of the y-axis is a reflection of the part to the left of it and vice versa. The arrows on the left and the right of the curve indicate that it extends indefinitely in both directions.
Graph . Select integers for x, starting with and ending with 3.
Example 3 Graphing an Equation Using the Point-Plotting Method
Graph . Select integers for x, starting with and ending with 3.
Solution
For each value of x, we find the corresponding value for y.
x
Ordered Pair
0
1
2
3
We plot the points and connect them, resulting in the graph shown in Figure 1.5. The graph is V-shaped and centered at the origin. For every point on the graph, the point is also on the graph. This shows that the absolute value of a positive number is the same as the absolute value of its opposite.
Graph . Select integers for x, starting with and ending with 2.
Objective 2: Graph equations in the rectangular coordinate system
Graphs of Equations
Objective 2 Graph equations in the rectangular coordinate system.
Watch Video
A relationship between two quantities can be expressed as an equation in two variables, such as
A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement. For example, consider the equation and the ordered pair . When 3 is substituted for x and is substituted for y, we obtain the statement , or , or . Because this statement is true, the ordered pair is a solution of the equation . We also say that satisfies the equation.
We can generate as many ordered-pair solutions as desired to by substituting numbers for x and then finding the corresponding values for y. For example, suppose we let
The graph of an equation in two variables is the set of all points whose coordinates satisfy the equation. One method for graphing such equations is the point-plotting method. First, we find several ordered pairs that are solutions of the equation. Next, we plot these ordered pairs as points in the rectangular coordinate system. Finally, we connect the points with a smooth curve or line. This often gives us a picture of all ordered pairs that satisfy the equation.
Example 2 Graphing an Equation Using the Point-Plotting Method
Graph . Select integers for x, starting with and ending with 3.
Solution
For each value of x, we find the corresponding value for y.
Now we plot the seven points and join them with a smooth curve, as shown in Figure 1.4. The graph of is a curve where the part of the graph to the right of the y-axis is a reflection of the part to the left of it and vice versa. The arrows on the left and the right of the curve indicate that it extends indefinitely in both directions.
Graph . Select integers for x, starting with and ending with 3.
Example 3 Graphing an Equation Using the Point-Plotting Method
Graph . Select integers for x, starting with and ending with 3.
Solution
For each value of x, we find the corresponding value for y.
x
Ordered Pair
0
1
2
3
We plot the points and connect them, resulting in the graph shown in Figure 1.5. The graph is V-shaped and centered at the origin. For every point on the graph, the point is also on the graph. This shows that the absolute value of a positive number is the same as the absolute value of its opposite.
Graph . Select integers for x, starting with and ending with 2.
Objective 3: Interpret information about a graphing utility’s viewing rectangle or table
Graphing Equations and Creating Tables Using a Graphing Utility
Objective 3 Interpret information about a graphing utility’s viewing rectangle or table.
Watch Video
Graphing calculators and graphing software packages for computers are referred to as graphing utilities or graphers. A graphing utility is a powerful tool that quickly generates the graph of an equation in two variables. Figures 1.6(a) and 1.6(b) show two such graphs for the equations in Examples 2 and 3.
What differences do you notice between these graphs and the graphs that we drew by hand? They do seem a bit “jittery.” Arrows do not appear on the left and right ends of the graphs. Furthermore, numbers are not given along the axes. For both graphs in Figure 1.6, the x-axis extends from to 10 and the y-axis also extends from to 10. The distance represented by each consecutive tick mark is one unit. We say that the viewing rectangle, or the viewing window, is by .
To graph an equation in x and y using a graphing utility, enter the equation and specify the size of the viewing rectangle. The size of the viewing rectangle sets minimum and maximum values for both the x- and y-axes. Enter these values, as well as the values representing the distances between consecutive tick marks, on the respective axes. The by viewing rectangle used in Figure 1.6 is called the standard viewing rectangle.
Example 4 Understanding the Viewing Rectangle
What is the meaning of a by viewing rectangle?
Solution
We begin with , which describes the x-axis. The minimum x-value is and the maximum x-value is 3. The distance between consecutive tick marks is 0.5.
Next, consider , which describes the y-axis. The minimum y-value is and the maximum y-value is 20. The distance between consecutive tick marks is 5.
Figure 1.7 illustrates a by viewing rectangle. To make things clearer, we’ve placed numbers by each tick mark. These numbers do not appear on the axes when you use a graphing utility to graph an equation.
What is the meaning of a by viewing rectangle? Create a figure like the one in Figure 1.7 that illustrates this viewing rectangle.
On many graphing utilities, the display screen is five-eighths as high as it is wide. By using a square setting, you can equally space the x and y tick marks. (This does not occur in the standard viewing rectangle.) Graphing utilities can also zoom in and zoom out. When you zoom in, you see a smaller portion of the graph, but you do so in greater detail. When you zoom out, you see a larger portion of the graph. Thus, zooming out may help you to develop a better understanding of the overall character of the graph. With practice, you will become more comfortable with graphing equations in two variables using your graphing utility. You will also develop a better sense of the size of the viewing rectangle that will reveal needed information about a particular graph.
Graphing utilities can also be used to create tables showing solutions of equations in two variables. Use the Table Setup function to choose the starting value of x and to input the increment, or change, between the consecutive x-values. The corresponding y-values are calculated based on the equation(s) in two variables in the screen. In Figure 1.8, we used a TI-84 Plus C to create a table for and , the equations in Examples 2 and 3.
An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis. For example, look at the graph of in Figure 1.9 at the top of the next page. The graph crosses the x-axis at and . Thus, the x-intercepts are and 2. They-coordinatecorresponding to an x-intercept is always zero.
A y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. The graph of in Figure 1.9 shows that the graph crosses the y-axis at . Thus, the y-intercept is 4. The x-coordinate corresponding to a y-intercept is always zero.
The graph crosses the x-axis at . Thus, the x-intercept is . The graph crosses the y-axis at . Thus, the y-intercept is 2.
The graph crosses the x-axis at , so the x-intercept is 3. This vertical line does not cross the y-axis. Thus, there is no y-intercept.
This graph crosses the x- and y-axes at the same point, the origin. Because the graph crosses both axes at , the x-intercept is 0 and the y-intercept is 0.
The graph crosses the x-axis at and . Thus, the x-intercepts are and 2. The graph crosses the y-axis at . Thus, the y-intercept is .
An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis. For example, look at the graph of in Figure 1.9 at the top of the next page. The graph crosses the x-axis at and . Thus, the x-intercepts are and 2. They-coordinatecorresponding to an x-intercept is always zero.
A y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. The graph of in Figure 1.9 shows that the graph crosses the y-axis at . Thus, the y-intercept is 4. The x-coordinate corresponding to a y-intercept is always zero.
The graph crosses the x-axis at . Thus, the x-intercept is . The graph crosses the y-axis at . Thus, the y-intercept is 2.
The graph crosses the x-axis at , so the x-intercept is 3. This vertical line does not cross the y-axis. Thus, there is no y-intercept.
This graph crosses the x- and y-axes at the same point, the origin. Because the graph crosses both axes at , the x-intercept is 0 and the y-intercept is 0.
The graph crosses the x-axis at and . Thus, the x-intercepts are and 2. The graph crosses the y-axis at . Thus, the y-intercept is .
Objective 5: Interpret information given by graphs
Interpreting Information Given by Graphs
Objective 5 Interpret information given by graphs.
Watch Video
Line graphs are often used to illustrate trends over time. Some measure of time, such as months or years, frequently appears on the horizontal axis. Amounts are generally listed on the vertical axis. Points are drawn to represent the given information. The graph is formed by connecting the points with line segments.
A line graph displays information in the first quadrant of a rectangular coordinate system. By identifying points on line graphs and their coordinates, you can interpret specific information given by the graph.
Example 6 Educational Attainment and the Probability of Divorce
Many factors affect whether a marriage will last. Data show that divorce rates are higher for those who marry younger, but education may also play a role in the longevity of a marriage. The line graphs in Figure 1.11 show the percentages of marriages ending in divorce after 5, 10, and 15 years of marriage for two levels of educational attainment.
Here are two mathematical models that approximate the data displayed by the line graphs:
In each model, the variable n is the number of years after marriage and the variable d is the percentage of marriages ending in divorce.
Use the appropriate formula to determine the percentage of marriages ending in divorce after 10 years when the educational attainment is bachelor’s degree or higher.
Use the appropriate line graph in Figure 1.11 to determine the percentage of marriages ending in divorce after 10 years when the educational attainment is bachelor’s degree or higher.
Does the value given by the mathematical model underestimate or overestimate the actual percentage of marriages ending in divorce after 10 years as shown by the graph? By how much?
Solution
Because the educational attainment is bachelor’s degree or higher, we use the formula on the right, . To find the percentage of marriages ending in divorce after 10 years, we substitute 10 for n and evaluate the formula.
The model indicates that 19% of marriages end in divorce after 10 years for those with a bachelor’s degree or higher.
Now let’s use the line graph that shows the percentage of marriages ending in divorce for a bachelor’s degree or higher. The graph is shown again in Figure 1.12. To find the percentage of marriages ending in divorce after 10 years:
Locate 10 on the horizontal axis and locate the point above 10.
Read across to the corresponding percent on the vertical axis.
The actual data displayed by the graph indicate that 20% of these marriages end in divorce after 10 years.
The value obtained by evaluating the mathematical model, 19%, is close to, but slightly less than, the actual percentage of divorces, 20%. The difference between these percents is , or 1%. The value given by the mathematical model, 19%, underestimates the actual percent, 20%, by only 1, providing a fairly accurate description of the data.
The data presented in Example 6 indicate longer-lasting marriages for those with more education. Does this mean that a college education prepares you for marriage? Not necessarily. Although the academics and social interactions in college may broaden a person’s perspective of relationships, there are underlying issues at play here. One such issue is finances: More education typically means higher earnings, which may lead to greater financial stability. Financial stress in a marriage is a major contributor to divorce.
Throughout this text, you will encounter models based on real-world data. To keep our models manageable, we limit the number of factors under consideration. However, we want you to be aware of the existence of underlying issues that may offer an alternative explanation for the trends observed in our numerous graphs and models.
Our goal is that you acquire a greater appreciation of mathematical models and their applications both in other academic disciplines and in real life, but we also want you to be aware of the limitations of these models.
Use the appropriate formula from Example 6 to determine the percentage of marriages ending in divorce after 15 years for high school graduates with no college.
Use the appropriate line graph in Figure 1.11 to determine the percentage of marriages ending in divorce after 15 years for high school graduates with no college.
Does the value given by the mathematical model underestimate or overestimate the actual percentage of marriages ending in divorce after 15 years as shown by the graph? By how much?
Objective 5: Interpret information given by graphs
Interpreting Information Given by Graphs
Objective 5 Interpret information given by graphs.
Watch Video
Line graphs are often used to illustrate trends over time. Some measure of time, such as months or years, frequently appears on the horizontal axis. Amounts are generally listed on the vertical axis. Points are drawn to represent the given information. The graph is formed by connecting the points with line segments.
A line graph displays information in the first quadrant of a rectangular coordinate system. By identifying points on line graphs and their coordinates, you can interpret specific information given by the graph.
Example 6 Educational Attainment and the Probability of Divorce
Many factors affect whether a marriage will last. Data show that divorce rates are higher for those who marry younger, but education may also play a role in the longevity of a marriage. The line graphs in Figure 1.11 show the percentages of marriages ending in divorce after 5, 10, and 15 years of marriage for two levels of educational attainment.
Here are two mathematical models that approximate the data displayed by the line graphs:
In each model, the variable n is the number of years after marriage and the variable d is the percentage of marriages ending in divorce.
Use the appropriate formula to determine the percentage of marriages ending in divorce after 10 years when the educational attainment is bachelor’s degree or higher.
Use the appropriate line graph in Figure 1.11 to determine the percentage of marriages ending in divorce after 10 years when the educational attainment is bachelor’s degree or higher.
Does the value given by the mathematical model underestimate or overestimate the actual percentage of marriages ending in divorce after 10 years as shown by the graph? By how much?
Solution
Because the educational attainment is bachelor’s degree or higher, we use the formula on the right, . To find the percentage of marriages ending in divorce after 10 years, we substitute 10 for n and evaluate the formula.
The model indicates that 19% of marriages end in divorce after 10 years for those with a bachelor’s degree or higher.
Now let’s use the line graph that shows the percentage of marriages ending in divorce for a bachelor’s degree or higher. The graph is shown again in Figure 1.12. To find the percentage of marriages ending in divorce after 10 years:
Locate 10 on the horizontal axis and locate the point above 10.
Read across to the corresponding percent on the vertical axis.
The actual data displayed by the graph indicate that 20% of these marriages end in divorce after 10 years.
The value obtained by evaluating the mathematical model, 19%, is close to, but slightly less than, the actual percentage of divorces, 20%. The difference between these percents is , or 1%. The value given by the mathematical model, 19%, underestimates the actual percent, 20%, by only 1, providing a fairly accurate description of the data.
The data presented in Example 6 indicate longer-lasting marriages for those with more education. Does this mean that a college education prepares you for marriage? Not necessarily. Although the academics and social interactions in college may broaden a person’s perspective of relationships, there are underlying issues at play here. One such issue is finances: More education typically means higher earnings, which may lead to greater financial stability. Financial stress in a marriage is a major contributor to divorce.
Throughout this text, you will encounter models based on real-world data. To keep our models manageable, we limit the number of factors under consideration. However, we want you to be aware of the existence of underlying issues that may offer an alternative explanation for the trends observed in our numerous graphs and models.
Our goal is that you acquire a greater appreciation of mathematical models and their applications both in other academic disciplines and in real life, but we also want you to be aware of the limitations of these models.
Use the appropriate formula from Example 6 to determine the percentage of marriages ending in divorce after 15 years for high school graduates with no college.
Use the appropriate line graph in Figure 1.11 to determine the percentage of marriages ending in divorce after 15 years for high school graduates with no college.
Does the value given by the mathematical model underestimate or overestimate the actual percentage of marriages ending in divorce after 15 years as shown by the graph? By how much?
1.1: Exercise Set
1.1 Exercise Set
Practice Exercises
In Exercises 1–12, plot the given point in a rectangular coordinate system.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Graph each equation in Exercises 13–28. Let , and 3.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
In Exercises 29–32, match the viewing rectangle with the correct figure. Then label the tick marks in the figure to illustrate this viewing rectangle.
29. by
30. by
31. by
32. by
The table of values was generated by a graphing utility with a TABLE feature. Use the table to solve Exercises 33–40.
37. At which point does the graph of cross the x-axis?
38. At which point does the graph of cross the y-axis?
39. At which points do the graphs of and intersect?
40. For which values of x is
In Exercises 41–46, use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.
The data can be described by the following mathematical models:
Use this information to solve Exercises 55–56.
55.
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2010. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2010. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was marijuana use by seniors at a maximum? Estimate the percentage of seniors who had ever used marijuana in that year.
56.
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2015. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2015. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was alcohol use by seniors at a maximum? What percentage of seniors had ever used alcohol in that year?
Contrary to popular belief, older people do not need less sleep than younger adults. However, the line graphs show that they awaken more often during the night. The numerous awakenings are one reason why some elderly individuals report that sleep is less restful than it had been in the past. Use the line graphs to solve Exercises 57–60.
Source: Stephen Davis and Joseph Palladino, Psychology, 5th Edition, Prentice Hall, 2007
57. At which age, estimated to the nearest year, do women have the least number of awakenings during the night? What is the average number of awakenings at that age?
58. At which age do men have the greatest number of awakenings during the night? What is the average number of awakenings at that age?
59. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 25-year-old men and 25-year-old women.
60. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 18-year-old men and 18-year-old women.
Explaining the Concepts
61. What is the rectangular coordinate system?
62. Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.
63. Explain why and do not represent the same point.
64. Explain how to graph an equation in the rectangular coordinate system.
65. What does a by viewing rectangle mean?
Technology Exercise
66. Use a graphing utility to verify each of your hand-drawn graphs in Exercises 13–28. Experiment with the settings for the viewing rectangle to make the graph displayed by the graphing utility resemble your hand-drawn graph as much as possible.
Critical Thinking Exercises
MAKE SENSE? In Exercises 67–70, determine whether each statement makes sense or does not make sense, and explain your reasoning.
67. The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like.
68. There is something wrong with my graphing utility because it is not displaying numbers along the x- and y-axes.
69. I used the ordered pairs , , and to graph a straight line.
70. I used the ordered pairs
(time of day, calories that I burned)
to obtain a graph that is a horizontal line.
In Exercises 71–74, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
71. If the product of a point’s coordinates is positive, the point must be in quadrant I.
72. If a point is on the x-axis, it is neither up nor down, so .
73. If a point is on the y-axis, its x-coordinate must be 0.
74. The ordered pair satisfies .
In Exercises 75–78, list the quadrant or quadrants satisfying each condition.
75.
76.
77.
78.
In Exercises 79–82, match the story with the correct figure. The figures are labeled (a), (b), (c), and (d).
79. As the blizzard got worse, the snow fell harder and harder.
80. The snow fell more and more softly.
81. It snowed hard, but then it stopped. After a short time, the snow started falling softly.
82. It snowed softly, and then it stopped. After a short time, the snow started falling hard.
In Exercises 83–87, select the graph that best illustrates each story.
83. An airplane flew from Miami to San Francisco. [Graphs (c) and (d) are at the top of the next page.]
84. At noon, you begin to breathe in.
85. Measurements are taken of a person’s height from birth to age 100.
86. You begin your bike ride by riding down a hill. Then you ride up another hill. Finally, you ride along a level surface before coming to a stop. [Graphs (c) and (d) are at the top of the next column.]
87. In Example 6, we used the formula to model the percentage of marriages ending in divorce, d, after n years of marriage for an educational attainment of bachelor’s degree or higher. We can also model the data with the formula
Using a calculator, evaluate each formula for , and 15. Round to the nearest tenth, where necessary. Which model appears to give the better estimates of the percentages shown in Figure 1.11?
Group Exercise
88. The group should identify three free online graphing calculators. Each group member should graph five equations from this exercise set using each of the three graphing calculators. Then, as a group, discuss what you like or don’t like about the calculators. Based on this discussion, make a list of criteria that you would use in choosing a graphing calculator. Which, if any, of the three graphing calculators would you recommend to fellow students?
Preview Exercises
Exercises 89–91 will help you prepare for the material covered in the next section.
89. Here are two sets of ordered pairs:
In which set is each x-coordinate paired with only one y-coordinate?
90. Graph and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2.
91. Use the following graph to solve this exercise.
37. At which point does the graph of cross the x-axis?
38. At which point does the graph of cross the y-axis?
39. At which points do the graphs of and intersect?
40. For which values of x is
In Exercises 41–46, use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.
The data can be described by the following mathematical models:
Use this information to solve Exercises 55–56.
55.
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2010. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2010. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was marijuana use by seniors at a maximum? Estimate the percentage of seniors who had ever used marijuana in that year.
56.
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2015. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2015. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was alcohol use by seniors at a maximum? What percentage of seniors had ever used alcohol in that year?
Contrary to popular belief, older people do not need less sleep than younger adults. However, the line graphs show that they awaken more often during the night. The numerous awakenings are one reason why some elderly individuals report that sleep is less restful than it had been in the past. Use the line graphs to solve Exercises 57–60.
Source: Stephen Davis and Joseph Palladino, Psychology, 5th Edition, Prentice Hall, 2007
57. At which age, estimated to the nearest year, do women have the least number of awakenings during the night? What is the average number of awakenings at that age?
58. At which age do men have the greatest number of awakenings during the night? What is the average number of awakenings at that age?
59. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 25-year-old men and 25-year-old women.
60. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 18-year-old men and 18-year-old women.
Explaining the Concepts
61. What is the rectangular coordinate system?
62. Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.
63. Explain why and do not represent the same point.
64. Explain how to graph an equation in the rectangular coordinate system.
65. What does a by viewing rectangle mean?
Technology Exercise
66. Use a graphing utility to verify each of your hand-drawn graphs in Exercises 13–28. Experiment with the settings for the viewing rectangle to make the graph displayed by the graphing utility resemble your hand-drawn graph as much as possible.
Critical Thinking Exercises
MAKE SENSE? In Exercises 67–70, determine whether each statement makes sense or does not make sense, and explain your reasoning.
67. The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like.
68. There is something wrong with my graphing utility because it is not displaying numbers along the x- and y-axes.
69. I used the ordered pairs , , and to graph a straight line.
70. I used the ordered pairs
(time of day, calories that I burned)
to obtain a graph that is a horizontal line.
In Exercises 71–74, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
71. If the product of a point’s coordinates is positive, the point must be in quadrant I.
72. If a point is on the x-axis, it is neither up nor down, so .
73. If a point is on the y-axis, its x-coordinate must be 0.
74. The ordered pair satisfies .
In Exercises 75–78, list the quadrant or quadrants satisfying each condition.
75.
76.
77.
78.
In Exercises 79–82, match the story with the correct figure. The figures are labeled (a), (b), (c), and (d).
79. As the blizzard got worse, the snow fell harder and harder.
80. The snow fell more and more softly.
81. It snowed hard, but then it stopped. After a short time, the snow started falling softly.
82. It snowed softly, and then it stopped. After a short time, the snow started falling hard.
In Exercises 83–87, select the graph that best illustrates each story.
83. An airplane flew from Miami to San Francisco. [Graphs (c) and (d) are at the top of the next page.]
84. At noon, you begin to breathe in.
85. Measurements are taken of a person’s height from birth to age 100.
86. You begin your bike ride by riding down a hill. Then you ride up another hill. Finally, you ride along a level surface before coming to a stop. [Graphs (c) and (d) are at the top of the next column.]
87. In Example 6, we used the formula to model the percentage of marriages ending in divorce, d, after n years of marriage for an educational attainment of bachelor’s degree or higher. We can also model the data with the formula
Using a calculator, evaluate each formula for , and 15. Round to the nearest tenth, where necessary. Which model appears to give the better estimates of the percentages shown in Figure 1.11?
Group Exercise
88. The group should identify three free online graphing calculators. Each group member should graph five equations from this exercise set using each of the three graphing calculators. Then, as a group, discuss what you like or don’t like about the calculators. Based on this discussion, make a list of criteria that you would use in choosing a graphing calculator. Which, if any, of the three graphing calculators would you recommend to fellow students?
Preview Exercises
Exercises 89–91 will help you prepare for the material covered in the next section.
89. Here are two sets of ordered pairs:
In which set is each x-coordinate paired with only one y-coordinate?
90. Graph and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2.
91. Use the following graph to solve this exercise.
37. At which point does the graph of cross the x-axis?
38. At which point does the graph of cross the y-axis?
39. At which points do the graphs of and intersect?
40. For which values of x is
In Exercises 41–46, use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.
The data can be described by the following mathematical models:
Use this information to solve Exercises 55–56.
55.
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2010. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2010. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was marijuana use by seniors at a maximum? Estimate the percentage of seniors who had ever used marijuana in that year.
56.
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2015. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2015. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was alcohol use by seniors at a maximum? What percentage of seniors had ever used alcohol in that year?
Contrary to popular belief, older people do not need less sleep than younger adults. However, the line graphs show that they awaken more often during the night. The numerous awakenings are one reason why some elderly individuals report that sleep is less restful than it had been in the past. Use the line graphs to solve Exercises 57–60.
Source: Stephen Davis and Joseph Palladino, Psychology, 5th Edition, Prentice Hall, 2007
57. At which age, estimated to the nearest year, do women have the least number of awakenings during the night? What is the average number of awakenings at that age?
58. At which age do men have the greatest number of awakenings during the night? What is the average number of awakenings at that age?
59. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 25-year-old men and 25-year-old women.
60. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 18-year-old men and 18-year-old women.
Explaining the Concepts
61. What is the rectangular coordinate system?
62. Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.
63. Explain why and do not represent the same point.
64. Explain how to graph an equation in the rectangular coordinate system.
65. What does a by viewing rectangle mean?
Technology Exercise
66. Use a graphing utility to verify each of your hand-drawn graphs in Exercises 13–28. Experiment with the settings for the viewing rectangle to make the graph displayed by the graphing utility resemble your hand-drawn graph as much as possible.
Critical Thinking Exercises
MAKE SENSE? In Exercises 67–70, determine whether each statement makes sense or does not make sense, and explain your reasoning.
67. The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like.
68. There is something wrong with my graphing utility because it is not displaying numbers along the x- and y-axes.
69. I used the ordered pairs , , and to graph a straight line.
70. I used the ordered pairs
(time of day, calories that I burned)
to obtain a graph that is a horizontal line.
In Exercises 71–74, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
71. If the product of a point’s coordinates is positive, the point must be in quadrant I.
72. If a point is on the x-axis, it is neither up nor down, so .
73. If a point is on the y-axis, its x-coordinate must be 0.
74. The ordered pair satisfies .
In Exercises 75–78, list the quadrant or quadrants satisfying each condition.
75.
76.
77.
78.
In Exercises 79–82, match the story with the correct figure. The figures are labeled (a), (b), (c), and (d).
79. As the blizzard got worse, the snow fell harder and harder.
80. The snow fell more and more softly.
81. It snowed hard, but then it stopped. After a short time, the snow started falling softly.
82. It snowed softly, and then it stopped. After a short time, the snow started falling hard.
In Exercises 83–87, select the graph that best illustrates each story.
83. An airplane flew from Miami to San Francisco. [Graphs (c) and (d) are at the top of the next page.]
84. At noon, you begin to breathe in.
85. Measurements are taken of a person’s height from birth to age 100.
86. You begin your bike ride by riding down a hill. Then you ride up another hill. Finally, you ride along a level surface before coming to a stop. [Graphs (c) and (d) are at the top of the next column.]
87. In Example 6, we used the formula to model the percentage of marriages ending in divorce, d, after n years of marriage for an educational attainment of bachelor’s degree or higher. We can also model the data with the formula
Using a calculator, evaluate each formula for , and 15. Round to the nearest tenth, where necessary. Which model appears to give the better estimates of the percentages shown in Figure 1.11?
Group Exercise
88. The group should identify three free online graphing calculators. Each group member should graph five equations from this exercise set using each of the three graphing calculators. Then, as a group, discuss what you like or don’t like about the calculators. Based on this discussion, make a list of criteria that you would use in choosing a graphing calculator. Which, if any, of the three graphing calculators would you recommend to fellow students?
Preview Exercises
Exercises 89–91 will help you prepare for the material covered in the next section.
89. Here are two sets of ordered pairs:
In which set is each x-coordinate paired with only one y-coordinate?
90. Graph and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2.
91. Use the following graph to solve this exercise.
37. At which point does the graph of cross the x-axis?
38. At which point does the graph of cross the y-axis?
39. At which points do the graphs of and intersect?
40. For which values of x is
In Exercises 41–46, use the graph to a. determine the x-intercepts, if any; b. determine the y-intercepts, if any. For each graph, tick marks along the axes represent one unit each.
The data can be described by the following mathematical models:
Use this information to solve Exercises 55–56.
55.
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2010. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2010.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2010. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was marijuana use by seniors at a maximum? Estimate the percentage of seniors who had ever used marijuana in that year.
56.
Use the appropriate line graph to estimate the percentage of seniors who had ever used alcohol in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used alcohol in 2015. How does this compare with your estimate in part (a)?
Use the appropriate line graph to estimate the percentage of seniors who had ever used marijuana in 2015.
Use the appropriate formula to determine the percentage of seniors who had ever used marijuana in 2015. How does this compare with your estimate in part (c)?
For the period from 1990 through 2019, in which year was alcohol use by seniors at a maximum? What percentage of seniors had ever used alcohol in that year?
Contrary to popular belief, older people do not need less sleep than younger adults. However, the line graphs show that they awaken more often during the night. The numerous awakenings are one reason why some elderly individuals report that sleep is less restful than it had been in the past. Use the line graphs to solve Exercises 57–60.
Source: Stephen Davis and Joseph Palladino, Psychology, 5th Edition, Prentice Hall, 2007
57. At which age, estimated to the nearest year, do women have the least number of awakenings during the night? What is the average number of awakenings at that age?
58. At which age do men have the greatest number of awakenings during the night? What is the average number of awakenings at that age?
59. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 25-year-old men and 25-year-old women.
60. Estimate, to the nearest tenth, the difference between the average number of awakenings during the night for 18-year-old men and 18-year-old women.
Explaining the Concepts
61. What is the rectangular coordinate system?
62. Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.
63. Explain why and do not represent the same point.
64. Explain how to graph an equation in the rectangular coordinate system.
65. What does a by viewing rectangle mean?
Technology Exercise
66. Use a graphing utility to verify each of your hand-drawn graphs in Exercises 13–28. Experiment with the settings for the viewing rectangle to make the graph displayed by the graphing utility resemble your hand-drawn graph as much as possible.
Critical Thinking Exercises
MAKE SENSE? In Exercises 67–70, determine whether each statement makes sense or does not make sense, and explain your reasoning.
67. The rectangular coordinate system provides a geometric picture of what an equation in two variables looks like.
68. There is something wrong with my graphing utility because it is not displaying numbers along the x- and y-axes.
69. I used the ordered pairs , , and to graph a straight line.
70. I used the ordered pairs
(time of day, calories that I burned)
to obtain a graph that is a horizontal line.
In Exercises 71–74, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
71. If the product of a point’s coordinates is positive, the point must be in quadrant I.
72. If a point is on the x-axis, it is neither up nor down, so .
73. If a point is on the y-axis, its x-coordinate must be 0.
74. The ordered pair satisfies .
In Exercises 75–78, list the quadrant or quadrants satisfying each condition.
75.
76.
77.
78.
In Exercises 79–82, match the story with the correct figure. The figures are labeled (a), (b), (c), and (d).
79. As the blizzard got worse, the snow fell harder and harder.
80. The snow fell more and more softly.
81. It snowed hard, but then it stopped. After a short time, the snow started falling softly.
82. It snowed softly, and then it stopped. After a short time, the snow started falling hard.
In Exercises 83–87, select the graph that best illustrates each story.
83. An airplane flew from Miami to San Francisco. [Graphs (c) and (d) are at the top of the next page.]
84. At noon, you begin to breathe in.
85. Measurements are taken of a person’s height from birth to age 100.
86. You begin your bike ride by riding down a hill. Then you ride up another hill. Finally, you ride along a level surface before coming to a stop. [Graphs (c) and (d) are at the top of the next column.]
87. In Example 6, we used the formula to model the percentage of marriages ending in divorce, d, after n years of marriage for an educational attainment of bachelor’s degree or higher. We can also model the data with the formula
Using a calculator, evaluate each formula for , and 15. Round to the nearest tenth, where necessary. Which model appears to give the better estimates of the percentages shown in Figure 1.11?
Group Exercise
88. The group should identify three free online graphing calculators. Each group member should graph five equations from this exercise set using each of the three graphing calculators. Then, as a group, discuss what you like or don’t like about the calculators. Based on this discussion, make a list of criteria that you would use in choosing a graphing calculator. Which, if any, of the three graphing calculators would you recommend to fellow students?
Preview Exercises
Exercises 89–91 will help you prepare for the material covered in the next section.
89. Here are two sets of ordered pairs:
In which set is each x-coordinate paired with only one y-coordinate?
90. Graph and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2.
91. Use the following graph to solve this exercise.
The number of T cells in a person with HIV is a function of time after infection. In this section, you will be introduced to the basics of functions and their graphs. We will analyze the graph of a function using an example that illustrates the progression of HIV and T cell count. Much of our work in this course will be devoted to the important topic of functions and how they model your world.
Magnified 6000 times, this color-scanned image shows a T-lymphocyte blood cell (green) infected with the HIV virus (red). Depletion of the number of T cells causes destruction of the immune system.
Objective 1: Find the domain and range of a relation
Relations
Objective 1 Find the domain and range of a relation.
Watch Video
Forbes magazine published a list of the highest-paid TV actors and actresses in 2018. The results are shown in Figure 1.13.
The graph indicates a correspondence between a TV actor or actress and that person’s earnings, in millions of dollars. We can write this correspondence using a set of ordered pairs:
The mathematical term for a set of ordered pairs is a relation.
Definition of a Relation
A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.
Example 1 Finding the Domain and Range of a Relation
Find the domain and range of the relation:
Solution
The domain is the set of all first components. Thus, the domain is
The range is the set of all second components. Thus, the range is
As you worked Check Point 1, did you wonder if there was a rule that assigned the “inputs” in the domain to the “outputs” in the range? For example, for the ordered pair (2, 225), how does the output 225 depend on the input 2? The ordered pair is based on the data in Figure 1.14(a), which shows the number of smartphone users, in millions, in the United States.
The graph indicates a correspondence between a TV actor or actress and that person’s earnings, in millions of dollars. We can write this correspondence using a set of ordered pairs:
The mathematical term for a set of ordered pairs is a relation.
Definition of a Relation
A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.
Example 1 Finding the Domain and Range of a Relation
Find the domain and range of the relation:
Solution
The domain is the set of all first components. Thus, the domain is
The range is the set of all second components. Thus, the range is
As you worked Check Point 1, did you wonder if there was a rule that assigned the “inputs” in the domain to the “outputs” in the range? For example, for the ordered pair (2, 225), how does the output 225 depend on the input 2? The ordered pair is based on the data in Figure 1.14(a), which shows the number of smartphone users, in millions, in the United States.
In Figure 1.14(b), we used the data for the number of smartphone users to create the following ordered pairs:
Consider, for example, the ordered pair (2, 225).
The four points in Figure 1.14(b) visually represent the relation formed from the data. Another way to visually represent the relation is as follows:
Objective 2: Determine whether a relation is a function
Functions
Objective 2 Determine whether a relation is a function.
Watch Video
Table 1.1, based on our earlier discussion, shows the highest-paid TV actors and actresses and their earnings in 2018, in millions of dollars. We’ve used this information to define two relations.
Table 1.1 Highest-Paid TV Actors and Actresses
Actor/Actress
Earnings (millions of dollars)
Vergara
43
Parsons
28
Galecki
27
Cuoco
26
Helberg
26
Figure 1.15(a) shows a correspondence between actors and actresses and their earnings. Figure 1.15(b) shows a correspondence between earnings and actors and actresses.
Figure 1.15(a) Actors and actresses correspond to earnings.
A relation in which each member of the domain corresponds to exactly one member of the range is a function. Can you see that the relation in Figure 1.15(a) is a function? Each actor or actress in the domain corresponds to exactly one earnings amount in the range. If we know the actor or actress, we can be sure of his or her earnings. Notice that more than one element in the domain can correspond to the same element in the range: Cuoco and Helberg both earned $26 million.
Is the relation in Figure 1.15(b) a function? Does each member of the domain correspond to precisely one member of the range? This relation is not a function because there is a member of the domain that corresponds to two different members of the range:
The member of the domain 26 corresponds to both Cuoco and Helberg. If we know that the earnings are $26 million, we cannot be sure of the actor or actress. Because a function is a relation in which no two ordered pairs have the same first component and different second components, the ordered pairs (26, Cuoco) and (26, Helberg) are not ordered pairs of a function.
Definition of a Function
A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range.
Example 2 illustrates that not every correspondence between sets is a function.
Example 2 Determining Whether a Relation Is a Function
Determine whether each relation is a function:
.
Solution
We begin by making a figure for each relation that shows the domain and the range (Figure 1.16).
Figure 1.16(a)
Figure 1.16(b)
Figure 1.16(a) shows that every element in the domain corresponds to exactly one element in the range. The element 1 in the domain corresponds to the element 6 in the range. Furthermore, 2 corresponds to 6, 3 corresponds to 8, and 4 corresponds to 9. No two ordered pairs in the given relation have the same first component and different second components. Thus, the relation is a function.
Figure 1.16(b) shows that 6 corresponds to both 1 and 2. If any element in the domain corresponds to more than one element in the range, the relation is not a function. This relation is not a function; two ordered pairs have the same first component and different second components.
Look at Figure 1.16(a) again. The fact that 1 and 2 in the domain correspond to the same number, 6, in the range does not violate the definition of a function. A function can have two different first components with the same second component. By contrast, a relation is not a function when two different ordered pairs have the same first component and different second components. Thus, the relation in Figure 1.16(b) is not a function.
Objective 3: Determine whether an equation represents a function
Functions as Equations
Objective 3 Determine whether an equation represents a function.
Watch Video
Functions are usually given in terms of equations rather than as sets of ordered pairs. For example, here is an equation that models the percentage of first-year college women claiming no religious affiliation as a function of time:
The variable x represents the number of years after 1970. The variable y represents the percentage of first-year college women claiming no religious affiliation. The variable y is a function of the variable x. For each value of x, there is one and only one value of y. The variable x is called the independent variable because it can be assigned any value from the domain. Thus, x can be assigned any nonnegative integer representing the number of years after 1970. The variable y is called the dependent variable because its value depends on x. The percentage claiming no religious affiliation depends on the number of years after 1970. The value of the dependent variable, y, is calculated after selecting a value for the independent variable, x.
We have seen that not every set of ordered pairs defines a function. Similarly, not all equations with the variables x and y define functions. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x.
Example 3 Determining Whether an Equation Represents a Function
Determine whether each equation defines y as a function of x:
.
Solution
Solve each equation for y in terms of x. If two or more values of y can be obtained for a given x, the equation is not a function.
From this last equation we can see that for each value of x, there is one and only one value of y. For example, if , then . The equation defines y as a function of x.
The in this last equation shows that for certain values of x (all values between and 2), there are two values of y. For example, if , then . For this reason, the equation does not define y as a function of x.
Solve each equation for y and then determine whether the equation defines y as a function of x:
.
Objective 4: Evaluate a function
Function Notation
Objective 4 Evaluate a function.
Watch Video
If an equation in x and y gives one and only one value of y for each value of x, then the variable y is a function of the variable x. When an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H. Any letter can be used to name a function. Suppose that f names a function. Think of the domain as the set of the function’s inputs and the range as the set of the function’s outputs. As shown in Figure 1.17, input is represented by x and the output by . The special notation , read “f of x” or “f at x,” represents the value of the function at the number x.
Figure 1.17 A “function machine” with inputs and outputs
Let’s make this clearer by considering a specific example. The equation models the percentage of college women claiming no religious affiliation, y, x years after 1970. This equation defines y as a function of x. We’ll name the function f. Now, we can apply our new function notation.
Suppose we are interested in finding , the function’s output when the input is 30. To find the value of the function at 30, we substitute 30 for x. We are evaluating the function at 30.
The statement , read “f of 30 equals 13.5,” tells us that the value of the function at 30 is 13.5. When the function’s input is 30, its output is 13.5. Figure 1.18 illustrates the input and output in terms of a function machine.
Figure 1.18 A function machine at work
In 2000, 13.2% actually claimed nonaffiliation, so our function that models the data slightly overestimates the percent for 2000.
We used to find . To find other function values, such as or , substitute the specified input value, 40 or 55, for x in the function’s equation.
If a function is named f and x represents the independent variable, the notation corresponds to the y-value for a given x. Thus,
define the same function. This function may be written as
Example 4 Evaluating a Function
If , evaluate each of the following:
.
Solution
We substitute , and for x in the equation for f. When replacing x with a variable or an algebraic expression, you might find it helpful to think of the function’s equation as
If an equation in x and y gives one and only one value of y for each value of x, then the variable y is a function of the variable x. When an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H. Any letter can be used to name a function. Suppose that f names a function. Think of the domain as the set of the function’s inputs and the range as the set of the function’s outputs. As shown in Figure 1.17, input is represented by x and the output by . The special notation , read “f of x” or “f at x,” represents the value of the function at the number x.
Figure 1.17 A “function machine” with inputs and outputs
Let’s make this clearer by considering a specific example. The equation models the percentage of college women claiming no religious affiliation, y, x years after 1970. This equation defines y as a function of x. We’ll name the function f. Now, we can apply our new function notation.
Suppose we are interested in finding , the function’s output when the input is 30. To find the value of the function at 30, we substitute 30 for x. We are evaluating the function at 30.
The statement , read “f of 30 equals 13.5,” tells us that the value of the function at 30 is 13.5. When the function’s input is 30, its output is 13.5. Figure 1.18 illustrates the input and output in terms of a function machine.
Figure 1.18 A function machine at work
In 2000, 13.2% actually claimed nonaffiliation, so our function that models the data slightly overestimates the percent for 2000.
We used to find . To find other function values, such as or , substitute the specified input value, 40 or 55, for x in the function’s equation.
If a function is named f and x represents the independent variable, the notation corresponds to the y-value for a given x. Thus,
define the same function. This function may be written as
Example 4 Evaluating a Function
If , evaluate each of the following:
.
Solution
We substitute , and for x in the equation for f. When replacing x with a variable or an algebraic expression, you might find it helpful to think of the function’s equation as
The graph of a function is the graph of its ordered pairs. For example, the graph of is the set of points in the rectangular coordinate system satisfying . Similarly, the graph of is the set of points in the rectangular coordinate system satisfying the equation . In the next example, we graph both of these functions in the same rectangular coordinate system.
Example 5 Graphing Functions
Graph the functions and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2.
Solution
We begin by setting up a partial table of coordinates for each function. Then we plot the five points in each table and connect them, as shown in Figure 1.19 on the next page. The graph of each function is a straight line. Do you see a relationship between the two graphs? The graph of g is the graph of f shifted vertically up by 4 units.
The graphs in Example 5 are straight lines. All functions with equations of the form graph as straight lines. Such functions, called linear functions, will be discussed in detail in Section 1.4.
Graph the functions and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2. How is the graph of g related to the graph of f?
The Vertical Line Test
Not every graph in the rectangular coordinate system is the graph of a function. The definition of a function specifies that no value of x can be paired with two or more different values of y. Consequently, if a graph contains two or more different points with the same first coordinate, the graph cannot represent a function. This is illustrated in Figure 1.20. Observe that points sharing a common first coordinate are vertically above or below each other.
Figure 1.20y is not a function of x because 0 is paired with three values of y, namely, 1, 0, and .
This observation is the basis of a useful test for determining whether a graph defines y as a function of x. The test is called the vertical line test.
Objective 5: Graph functions by plotting points
Graphs of Functions
Objective 5 Graph functions by plotting points.
Watch Video
The graph of a function is the graph of its ordered pairs. For example, the graph of is the set of points in the rectangular coordinate system satisfying . Similarly, the graph of is the set of points in the rectangular coordinate system satisfying the equation . In the next example, we graph both of these functions in the same rectangular coordinate system.
Example 5 Graphing Functions
Graph the functions and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2.
Solution
We begin by setting up a partial table of coordinates for each function. Then we plot the five points in each table and connect them, as shown in Figure 1.19 on the next page. The graph of each function is a straight line. Do you see a relationship between the two graphs? The graph of g is the graph of f shifted vertically up by 4 units.
The graphs in Example 5 are straight lines. All functions with equations of the form graph as straight lines. Such functions, called linear functions, will be discussed in detail in Section 1.4.
Graph the functions and in the same rectangular coordinate system. Select integers for x, starting with and ending with 2. How is the graph of g related to the graph of f?
The Vertical Line Test
Not every graph in the rectangular coordinate system is the graph of a function. The definition of a function specifies that no value of x can be paired with two or more different values of y. Consequently, if a graph contains two or more different points with the same first coordinate, the graph cannot represent a function. This is illustrated in Figure 1.20. Observe that points sharing a common first coordinate are vertically above or below each other.
Figure 1.20y is not a function of x because 0 is paired with three values of y, namely, 1, 0, and .
This observation is the basis of a useful test for determining whether a graph defines y as a function of x. The test is called the vertical line test.
Objective 7: Obtain information about a function from its graph
Obtaining Information from Graphs
Objective 7 Obtain information about a function from its graph.
Watch Video
You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows.
A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph.
An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph.
An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.
Example 7 Analyzing the Graph of a Function
The human immunodeficiency virus, or HIV, infects and kills helper T cells. Because T cells stimulate the immune system to produce antibodies, their destruction disables the body’s defenses against other pathogens. By counting the number of T cells that remain active in the body, the progression of HIV can be monitored. The fewer helper T cells, the more advanced the disease. Without the drugs that are now used to inhibit the progression of the virus, Figure 1.21 shows a graph that is used to monitor the average progression of the disease. The number of T cells, , is a function of time after infection, x.
Figure 1.21
Source: B. E. Pruitt et al., Human Sexuality, Prentice Hall, 2007
No vertical line can be drawn that intersects the graph of f more than once. By the vertical line test, f represents the graph of a function.
To find , or f of 8, we locate 8 on the x-axis. Figure 1.22 shows the point on the graph of f for which 8 is the first coordinate. From this point, we look to the y-axis to find the corresponding y-coordinate. We see that the y-coordinate is 200. Thus,
When the time after infection is 8 years, the T cell count is 200 cells per milliliter of blood. (AIDS clinical diagnosis is given at a T cell count of 200 or below.)
To find the value of x for which , we find the approximate location of 350 on the y-axis. Figure 1.23 shows that there is one point on the graph of f for which 350 is the second coordinate. From this point, we look to the x-axis to find the corresponding x-coordinate. We see that the x-coordinate is 6. Thus,
Estimate the minimum T cell count during the asymptomatic stage.
Objective 7: Obtain information about a function from its graph
Obtaining Information from Graphs
Objective 7 Obtain information about a function from its graph.
Watch Video
You can obtain information about a function from its graph. At the right or left of a graph, you will find closed dots, open dots, or arrows.
A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph.
An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph.
An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.
Example 7 Analyzing the Graph of a Function
The human immunodeficiency virus, or HIV, infects and kills helper T cells. Because T cells stimulate the immune system to produce antibodies, their destruction disables the body’s defenses against other pathogens. By counting the number of T cells that remain active in the body, the progression of HIV can be monitored. The fewer helper T cells, the more advanced the disease. Without the drugs that are now used to inhibit the progression of the virus, Figure 1.21 shows a graph that is used to monitor the average progression of the disease. The number of T cells, , is a function of time after infection, x.
Figure 1.21
Source: B. E. Pruitt et al., Human Sexuality, Prentice Hall, 2007
No vertical line can be drawn that intersects the graph of f more than once. By the vertical line test, f represents the graph of a function.
To find , or f of 8, we locate 8 on the x-axis. Figure 1.22 shows the point on the graph of f for which 8 is the first coordinate. From this point, we look to the y-axis to find the corresponding y-coordinate. We see that the y-coordinate is 200. Thus,
When the time after infection is 8 years, the T cell count is 200 cells per milliliter of blood. (AIDS clinical diagnosis is given at a T cell count of 200 or below.)
To find the value of x for which , we find the approximate location of 350 on the y-axis. Figure 1.23 shows that there is one point on the graph of f for which 350 is the second coordinate. From this point, we look to the x-axis to find the corresponding x-coordinate. We see that the x-coordinate is 6. Thus,
Let’s apply these ideas to the graph of the function shown in Figure 1.26. To find the domain, look for all the inputs on the x-axis that correspond to points on the graph. Can you see that they extend from to 2, inclusive? The function’s domain can be represented as follows:
To find the range, look for all the outputs on the y-axis that correspond to points on the graph. They extend from 1 to 4, inclusive. The function’s range can be represented as follows:
Let’s apply these ideas to the graph of the function shown in Figure 1.26. To find the domain, look for all the inputs on the x-axis that correspond to points on the graph. Can you see that they extend from to 2, inclusive? The function’s domain can be represented as follows:
To find the range, look for all the outputs on the y-axis that correspond to points on the graph. They extend from 1 to 4, inclusive. The function’s range can be represented as follows:
Objective 9: Identify intercepts from a function’s graph
Identifying Intercepts from a Function’s Graph
Objective 9 Identify intercepts from a function’s graph.
Watch Video
Figure 1.27 illustrates how we can identify intercepts from a function’s graph. To find the x-intercepts, look for the points at which the graph crosses the x-axis. There are three such points: , , and . Thus, the x-intercepts are , and 5. We express this in function notation by writing , and . We say that , and 5 are the zeros of the function. The zeros of a functionf are the for which . Thus, the real zeros are the x-intercepts.
To find the y-intercept, look for the point at which the graph crosses the y-axis. This occurs at . Thus, the y-intercept is 3. We express this in function notation by writing .
By the definition of a function, for each value of x we can have at most one value for y. What does this mean in terms of intercepts? A function can have more than one x-intercept but at most one y-intercept.
1.2: Exercise Set
1.2 Exercise Set
Practice Exercises
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–26, determine whether each equation defines y as a function of x.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
51. and
52. and
53. and
54. and
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–70, use the graph of f to find each indicated function value.
65.
66.
67.
68.
69.
70.
Use the graph of g to solve Exercises 71–76.
71. Find .
72. Find .
73. Find .
74. Find .
75. For what value of x is
76. For what value of x is
In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
In Exercises 95–96, let f and g be defined by the following table:
x
6
0
3
4
0
1
1
2
0
95. Find .
96. Find .
In Exercises 97–98, find for the given function f. Then simplify the expression.
97.
98.
Application Exercises
The bar graph shows public spending by the top five and the bottom five countries on pre-primary education and child care. Spending is given by public expenditure as a percentage of gross domestic product. Use the graph to solve Exercises 99–100.
Write a set of five ordered pairs in which each of the top-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five top-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
100.
Write a set of five ordered pairs in which each of the bottom-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five bottom-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
The bar graph shows your chances of surviving to various ages once you reach 60.
model the chance, as a percent, that a 60-year-old will survive to age x. Use this information to solve Exercises 101–102.
101.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 70?
102.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 90?
The wage gap, which is used to compare the status of women’s earnings relative to men’s, is expressed as a percent and is calculated by dividing the median, or middlemost, annual earnings for all women by the median annual earnings for all men. The bar graph shows the wage gap for selected years from 1980 through 2020.
The function models the wage gap, as a percent, x years after 1980. The graph of function G is shown to the right of the actual data. Use this information to solve Exercises 103–104.
103.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
104.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
In Exercises 105–108, you will be developing functions that model given conditions.
105. A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, C, as a function of the number of bicycles produced, x. Then find and interpret .
106. A previously owned car was purchased for $22,500. The value of the car decreased by $3200 per year for each of the next six years. Write a function that describes the value of the car, V, after x years, where . Then find and interpret .
107. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, T, in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip, x. Then find and interpret . Hint:
108. A chemist working on a flu vaccine needs to mix a 10% sodium-iodine solution with a 60% sodium-iodine solution to obtain a 50-milliliter mixture. Write the amount of sodium iodine in the mixture, S, in milliliters, as a function of the number of milliliters of the 10% solution used, x. Then find and interpret .
Explaining the Concepts
109. What is a relation? Describe what is meant by its domain and its range.
110. Explain how to determine whether a relation is a function. What is a function?
111. How do you determine if an equation in x and y defines y as a function of x?
112. Does mean f times x when referring to a function f? If not, what does mean? Provide an example with your explanation.
113. What is the graph of a function?
114. Explain how the vertical line test is used to determine whether a graph represents a function.
115. Explain how to identify the domain and range of a function from its graph.
116. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. By contrast, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices.
Describe an everyday situation between variables that is a function.
Describe an everyday situation between variables that is not a function.
Technology Exercise
117. Use a graphing utility to verify any five pairs of graphs that you drew by hand in Exercises 39–54.
Critical Thinking Exercises
MAKE SENSE? In Exercises 118–121, determine whether each statement makes sense or does not make sense, and explain your reasoning.
118. My body temperature is a function of the time of day.
119. Using , I found by applying the distributive property to .
120. I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.
121. I graphed a function showing how the average number of annual physician visits depends on a person’s age. The domain was the average number of annual physician visits.
Use the graph of f to determine whether each statement in Exercises 122–125 is true or false.
127. Give an example of a relation with the following characteristics: The relation is a function containing two ordered pairs. Reversing the components in each ordered pair results in a relation that is not a function.
131. Sharks may be scary, but they were responsible for only three deaths worldwide in 2014. The world’s deadliest creatures, ranked by the number of human deaths per year, are mosquitoes, snails, and snakes. The number of deaths by mosquitoes exceeds the number of deaths by snakes by 661 thousand. The number of deaths by snails exceeds the number of deaths by snakes by 106 thousand. Combined, mosquitoes, snails, and snakes result in 1049 thousand (or 1,049,000) human deaths per year. Determine the number of human deaths per year, in thousands, by snakes, mosquitoes, and snails. (Source: World Health Organization) (Section P.8, Example 1)
Preview Exercises
Exercises 132–134 will help you prepare for the material covered in the next section.
132. The function describes the monthly cost, , in dollars, for a high-speed wireless Internet plan for g gigabytes of data, where . Find and interpret C(45).
133. Use point plotting to graph if .
134. Simplify: .
1.2: Exercise Set
1.2 Exercise Set
Practice Exercises
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–26, determine whether each equation defines y as a function of x.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
51. and
52. and
53. and
54. and
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–70, use the graph of f to find each indicated function value.
65.
66.
67.
68.
69.
70.
Use the graph of g to solve Exercises 71–76.
71. Find .
72. Find .
73. Find .
74. Find .
75. For what value of x is
76. For what value of x is
In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
In Exercises 95–96, let f and g be defined by the following table:
x
6
0
3
4
0
1
1
2
0
95. Find .
96. Find .
In Exercises 97–98, find for the given function f. Then simplify the expression.
97.
98.
Application Exercises
The bar graph shows public spending by the top five and the bottom five countries on pre-primary education and child care. Spending is given by public expenditure as a percentage of gross domestic product. Use the graph to solve Exercises 99–100.
Write a set of five ordered pairs in which each of the top-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five top-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
100.
Write a set of five ordered pairs in which each of the bottom-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five bottom-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
The bar graph shows your chances of surviving to various ages once you reach 60.
model the chance, as a percent, that a 60-year-old will survive to age x. Use this information to solve Exercises 101–102.
101.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 70?
102.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 90?
The wage gap, which is used to compare the status of women’s earnings relative to men’s, is expressed as a percent and is calculated by dividing the median, or middlemost, annual earnings for all women by the median annual earnings for all men. The bar graph shows the wage gap for selected years from 1980 through 2020.
The function models the wage gap, as a percent, x years after 1980. The graph of function G is shown to the right of the actual data. Use this information to solve Exercises 103–104.
103.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
104.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
In Exercises 105–108, you will be developing functions that model given conditions.
105. A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, C, as a function of the number of bicycles produced, x. Then find and interpret .
106. A previously owned car was purchased for $22,500. The value of the car decreased by $3200 per year for each of the next six years. Write a function that describes the value of the car, V, after x years, where . Then find and interpret .
107. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, T, in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip, x. Then find and interpret . Hint:
108. A chemist working on a flu vaccine needs to mix a 10% sodium-iodine solution with a 60% sodium-iodine solution to obtain a 50-milliliter mixture. Write the amount of sodium iodine in the mixture, S, in milliliters, as a function of the number of milliliters of the 10% solution used, x. Then find and interpret .
Explaining the Concepts
109. What is a relation? Describe what is meant by its domain and its range.
110. Explain how to determine whether a relation is a function. What is a function?
111. How do you determine if an equation in x and y defines y as a function of x?
112. Does mean f times x when referring to a function f? If not, what does mean? Provide an example with your explanation.
113. What is the graph of a function?
114. Explain how the vertical line test is used to determine whether a graph represents a function.
115. Explain how to identify the domain and range of a function from its graph.
116. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. By contrast, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices.
Describe an everyday situation between variables that is a function.
Describe an everyday situation between variables that is not a function.
Technology Exercise
117. Use a graphing utility to verify any five pairs of graphs that you drew by hand in Exercises 39–54.
Critical Thinking Exercises
MAKE SENSE? In Exercises 118–121, determine whether each statement makes sense or does not make sense, and explain your reasoning.
118. My body temperature is a function of the time of day.
119. Using , I found by applying the distributive property to .
120. I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.
121. I graphed a function showing how the average number of annual physician visits depends on a person’s age. The domain was the average number of annual physician visits.
Use the graph of f to determine whether each statement in Exercises 122–125 is true or false.
127. Give an example of a relation with the following characteristics: The relation is a function containing two ordered pairs. Reversing the components in each ordered pair results in a relation that is not a function.
131. Sharks may be scary, but they were responsible for only three deaths worldwide in 2014. The world’s deadliest creatures, ranked by the number of human deaths per year, are mosquitoes, snails, and snakes. The number of deaths by mosquitoes exceeds the number of deaths by snakes by 661 thousand. The number of deaths by snails exceeds the number of deaths by snakes by 106 thousand. Combined, mosquitoes, snails, and snakes result in 1049 thousand (or 1,049,000) human deaths per year. Determine the number of human deaths per year, in thousands, by snakes, mosquitoes, and snails. (Source: World Health Organization) (Section P.8, Example 1)
Preview Exercises
Exercises 132–134 will help you prepare for the material covered in the next section.
132. The function describes the monthly cost, , in dollars, for a high-speed wireless Internet plan for g gigabytes of data, where . Find and interpret C(45).
133. Use point plotting to graph if .
134. Simplify: .
1.2: Exercise Set
1.2 Exercise Set
Practice Exercises
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–26, determine whether each equation defines y as a function of x.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
51. and
52. and
53. and
54. and
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–70, use the graph of f to find each indicated function value.
65.
66.
67.
68.
69.
70.
Use the graph of g to solve Exercises 71–76.
71. Find .
72. Find .
73. Find .
74. Find .
75. For what value of x is
76. For what value of x is
In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
In Exercises 95–96, let f and g be defined by the following table:
x
6
0
3
4
0
1
1
2
0
95. Find .
96. Find .
In Exercises 97–98, find for the given function f. Then simplify the expression.
97.
98.
Application Exercises
The bar graph shows public spending by the top five and the bottom five countries on pre-primary education and child care. Spending is given by public expenditure as a percentage of gross domestic product. Use the graph to solve Exercises 99–100.
Write a set of five ordered pairs in which each of the top-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five top-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
100.
Write a set of five ordered pairs in which each of the bottom-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five bottom-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
The bar graph shows your chances of surviving to various ages once you reach 60.
model the chance, as a percent, that a 60-year-old will survive to age x. Use this information to solve Exercises 101–102.
101.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 70?
102.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 90?
The wage gap, which is used to compare the status of women’s earnings relative to men’s, is expressed as a percent and is calculated by dividing the median, or middlemost, annual earnings for all women by the median annual earnings for all men. The bar graph shows the wage gap for selected years from 1980 through 2020.
The function models the wage gap, as a percent, x years after 1980. The graph of function G is shown to the right of the actual data. Use this information to solve Exercises 103–104.
103.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
104.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
In Exercises 105–108, you will be developing functions that model given conditions.
105. A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, C, as a function of the number of bicycles produced, x. Then find and interpret .
106. A previously owned car was purchased for $22,500. The value of the car decreased by $3200 per year for each of the next six years. Write a function that describes the value of the car, V, after x years, where . Then find and interpret .
107. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, T, in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip, x. Then find and interpret . Hint:
108. A chemist working on a flu vaccine needs to mix a 10% sodium-iodine solution with a 60% sodium-iodine solution to obtain a 50-milliliter mixture. Write the amount of sodium iodine in the mixture, S, in milliliters, as a function of the number of milliliters of the 10% solution used, x. Then find and interpret .
Explaining the Concepts
109. What is a relation? Describe what is meant by its domain and its range.
110. Explain how to determine whether a relation is a function. What is a function?
111. How do you determine if an equation in x and y defines y as a function of x?
112. Does mean f times x when referring to a function f? If not, what does mean? Provide an example with your explanation.
113. What is the graph of a function?
114. Explain how the vertical line test is used to determine whether a graph represents a function.
115. Explain how to identify the domain and range of a function from its graph.
116. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. By contrast, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices.
Describe an everyday situation between variables that is a function.
Describe an everyday situation between variables that is not a function.
Technology Exercise
117. Use a graphing utility to verify any five pairs of graphs that you drew by hand in Exercises 39–54.
Critical Thinking Exercises
MAKE SENSE? In Exercises 118–121, determine whether each statement makes sense or does not make sense, and explain your reasoning.
118. My body temperature is a function of the time of day.
119. Using , I found by applying the distributive property to .
120. I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.
121. I graphed a function showing how the average number of annual physician visits depends on a person’s age. The domain was the average number of annual physician visits.
Use the graph of f to determine whether each statement in Exercises 122–125 is true or false.
127. Give an example of a relation with the following characteristics: The relation is a function containing two ordered pairs. Reversing the components in each ordered pair results in a relation that is not a function.
131. Sharks may be scary, but they were responsible for only three deaths worldwide in 2014. The world’s deadliest creatures, ranked by the number of human deaths per year, are mosquitoes, snails, and snakes. The number of deaths by mosquitoes exceeds the number of deaths by snakes by 661 thousand. The number of deaths by snails exceeds the number of deaths by snakes by 106 thousand. Combined, mosquitoes, snails, and snakes result in 1049 thousand (or 1,049,000) human deaths per year. Determine the number of human deaths per year, in thousands, by snakes, mosquitoes, and snails. (Source: World Health Organization) (Section P.8, Example 1)
Preview Exercises
Exercises 132–134 will help you prepare for the material covered in the next section.
132. The function describes the monthly cost, , in dollars, for a high-speed wireless Internet plan for g gigabytes of data, where . Find and interpret C(45).
133. Use point plotting to graph if .
134. Simplify: .
1.2: Exercise Set
1.2 Exercise Set
Practice Exercises
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–26, determine whether each equation defines y as a function of x.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
51. and
52. and
53. and
54. and
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–70, use the graph of f to find each indicated function value.
65.
66.
67.
68.
69.
70.
Use the graph of g to solve Exercises 71–76.
71. Find .
72. Find .
73. Find .
74. Find .
75. For what value of x is
76. For what value of x is
In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
In Exercises 95–96, let f and g be defined by the following table:
x
6
0
3
4
0
1
1
2
0
95. Find .
96. Find .
In Exercises 97–98, find for the given function f. Then simplify the expression.
97.
98.
Application Exercises
The bar graph shows public spending by the top five and the bottom five countries on pre-primary education and child care. Spending is given by public expenditure as a percentage of gross domestic product. Use the graph to solve Exercises 99–100.
Write a set of five ordered pairs in which each of the top-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five top-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
100.
Write a set of five ordered pairs in which each of the bottom-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five bottom-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
The bar graph shows your chances of surviving to various ages once you reach 60.
model the chance, as a percent, that a 60-year-old will survive to age x. Use this information to solve Exercises 101–102.
101.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 70?
102.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 90?
The wage gap, which is used to compare the status of women’s earnings relative to men’s, is expressed as a percent and is calculated by dividing the median, or middlemost, annual earnings for all women by the median annual earnings for all men. The bar graph shows the wage gap for selected years from 1980 through 2020.
The function models the wage gap, as a percent, x years after 1980. The graph of function G is shown to the right of the actual data. Use this information to solve Exercises 103–104.
103.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
104.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
In Exercises 105–108, you will be developing functions that model given conditions.
105. A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, C, as a function of the number of bicycles produced, x. Then find and interpret .
106. A previously owned car was purchased for $22,500. The value of the car decreased by $3200 per year for each of the next six years. Write a function that describes the value of the car, V, after x years, where . Then find and interpret .
107. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, T, in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip, x. Then find and interpret . Hint:
108. A chemist working on a flu vaccine needs to mix a 10% sodium-iodine solution with a 60% sodium-iodine solution to obtain a 50-milliliter mixture. Write the amount of sodium iodine in the mixture, S, in milliliters, as a function of the number of milliliters of the 10% solution used, x. Then find and interpret .
Explaining the Concepts
109. What is a relation? Describe what is meant by its domain and its range.
110. Explain how to determine whether a relation is a function. What is a function?
111. How do you determine if an equation in x and y defines y as a function of x?
112. Does mean f times x when referring to a function f? If not, what does mean? Provide an example with your explanation.
113. What is the graph of a function?
114. Explain how the vertical line test is used to determine whether a graph represents a function.
115. Explain how to identify the domain and range of a function from its graph.
116. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. By contrast, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices.
Describe an everyday situation between variables that is a function.
Describe an everyday situation between variables that is not a function.
Technology Exercise
117. Use a graphing utility to verify any five pairs of graphs that you drew by hand in Exercises 39–54.
Critical Thinking Exercises
MAKE SENSE? In Exercises 118–121, determine whether each statement makes sense or does not make sense, and explain your reasoning.
118. My body temperature is a function of the time of day.
119. Using , I found by applying the distributive property to .
120. I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.
121. I graphed a function showing how the average number of annual physician visits depends on a person’s age. The domain was the average number of annual physician visits.
Use the graph of f to determine whether each statement in Exercises 122–125 is true or false.
127. Give an example of a relation with the following characteristics: The relation is a function containing two ordered pairs. Reversing the components in each ordered pair results in a relation that is not a function.
131. Sharks may be scary, but they were responsible for only three deaths worldwide in 2014. The world’s deadliest creatures, ranked by the number of human deaths per year, are mosquitoes, snails, and snakes. The number of deaths by mosquitoes exceeds the number of deaths by snakes by 661 thousand. The number of deaths by snails exceeds the number of deaths by snakes by 106 thousand. Combined, mosquitoes, snails, and snakes result in 1049 thousand (or 1,049,000) human deaths per year. Determine the number of human deaths per year, in thousands, by snakes, mosquitoes, and snails. (Source: World Health Organization) (Section P.8, Example 1)
Preview Exercises
Exercises 132–134 will help you prepare for the material covered in the next section.
132. The function describes the monthly cost, , in dollars, for a high-speed wireless Internet plan for g gigabytes of data, where . Find and interpret C(45).
133. Use point plotting to graph if .
134. Simplify: .
1.2: Exercise Set
1.2 Exercise Set
Practice Exercises
In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
In Exercises 11–26, determine whether each equation defines y as a function of x.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
In Exercises 51–54, graph the given square root functions, f and g, in the same rectangular coordinate system. Use the integer values of x given to the right of each function to obtain ordered pairs. Because only nonnegative numbers have square roots that are real numbers, be sure that each graph appears only for values of x that cause the expression under the radical sign to be greater than or equal to zero. Once you have obtained your graphs, describe how the graph of g is related to the graph of f.
51. and
52. and
53. and
54. and
In Exercises 55–64, use the vertical line test to identify graphs in which y is a function of x.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
In Exercises 65–70, use the graph of f to find each indicated function value.
65.
66.
67.
68.
69.
70.
Use the graph of g to solve Exercises 71–76.
71. Find .
72. Find .
73. Find .
74. Find .
75. For what value of x is
76. For what value of x is
In Exercises 77–92, use the graph to determine a. the function’s domain; b. the function’s range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
In Exercises 95–96, let f and g be defined by the following table:
x
6
0
3
4
0
1
1
2
0
95. Find .
96. Find .
In Exercises 97–98, find for the given function f. Then simplify the expression.
97.
98.
Application Exercises
The bar graph shows public spending by the top five and the bottom five countries on pre-primary education and child care. Spending is given by public expenditure as a percentage of gross domestic product. Use the graph to solve Exercises 99–100.
Write a set of five ordered pairs in which each of the top-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five top-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
100.
Write a set of five ordered pairs in which each of the bottom-spending countries corresponds to spending as a percentage of gross domestic product. Each ordered pair should be of the form
(country, percentage of gross domestic product).
Is the relation in part (a) a function? Explain your answer.
For the five bottom-spending countries, write a set of ordered pairs in which its spending as a percentage of gross domestic product corresponds to the country. Each ordered pair should be of the form
(percentage of gross domestic product, country).
Is the relation in part (c) a function? Explain your answer.
The bar graph shows your chances of surviving to various ages once you reach 60.
model the chance, as a percent, that a 60-year-old will survive to age x. Use this information to solve Exercises 101–102.
101.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 70?
102.
Find and interpret .
Find and interpret .
Which function serves as a better model for the chance of surviving to age 90?
The wage gap, which is used to compare the status of women’s earnings relative to men’s, is expressed as a percent and is calculated by dividing the median, or middlemost, annual earnings for all women by the median annual earnings for all men. The bar graph shows the wage gap for selected years from 1980 through 2020.
The function models the wage gap, as a percent, x years after 1980. The graph of function G is shown to the right of the actual data. Use this information to solve Exercises 103–104.
103.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
104.
Find and interpret . Identify this information as a point on the graph of the function.
Does overestimate or underestimate the actual data shown by the bar graph? By how much?
In Exercises 105–108, you will be developing functions that model given conditions.
105. A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, C, as a function of the number of bicycles produced, x. Then find and interpret .
106. A previously owned car was purchased for $22,500. The value of the car decreased by $3200 per year for each of the next six years. Write a function that describes the value of the car, V, after x years, where . Then find and interpret .
107. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, T, in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip, x. Then find and interpret . Hint:
108. A chemist working on a flu vaccine needs to mix a 10% sodium-iodine solution with a 60% sodium-iodine solution to obtain a 50-milliliter mixture. Write the amount of sodium iodine in the mixture, S, in milliliters, as a function of the number of milliliters of the 10% solution used, x. Then find and interpret .
Explaining the Concepts
109. What is a relation? Describe what is meant by its domain and its range.
110. Explain how to determine whether a relation is a function. What is a function?
111. How do you determine if an equation in x and y defines y as a function of x?
112. Does mean f times x when referring to a function f? If not, what does mean? Provide an example with your explanation.
113. What is the graph of a function?
114. Explain how the vertical line test is used to determine whether a graph represents a function.
115. Explain how to identify the domain and range of a function from its graph.
116. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. By contrast, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices.
Describe an everyday situation between variables that is a function.
Describe an everyday situation between variables that is not a function.
Technology Exercise
117. Use a graphing utility to verify any five pairs of graphs that you drew by hand in Exercises 39–54.
Critical Thinking Exercises
MAKE SENSE? In Exercises 118–121, determine whether each statement makes sense or does not make sense, and explain your reasoning.
118. My body temperature is a function of the time of day.
119. Using , I found by applying the distributive property to .
120. I graphed a function showing how paid vacation days depend on the number of years a person works for a company. The domain was the number of paid vacation days.
121. I graphed a function showing how the average number of annual physician visits depends on a person’s age. The domain was the average number of annual physician visits.
Use the graph of f to determine whether each statement in Exercises 122–125 is true or false.
127. Give an example of a relation with the following characteristics: The relation is a function containing two ordered pairs. Reversing the components in each ordered pair results in a relation that is not a function.
131. Sharks may be scary, but they were responsible for only three deaths worldwide in 2014. The world’s deadliest creatures, ranked by the number of human deaths per year, are mosquitoes, snails, and snakes. The number of deaths by mosquitoes exceeds the number of deaths by snakes by 661 thousand. The number of deaths by snails exceeds the number of deaths by snakes by 106 thousand. Combined, mosquitoes, snails, and snakes result in 1049 thousand (or 1,049,000) human deaths per year. Determine the number of human deaths per year, in thousands, by snakes, mosquitoes, and snails. (Source: World Health Organization) (Section P.8, Example 1)
Preview Exercises
Exercises 132–134 will help you prepare for the material covered in the next section.
132. The function describes the monthly cost, , in dollars, for a high-speed wireless Internet plan for g gigabytes of data, where . Find and interpret C(45).
It’s hard to believe that this gas-guzzler, with its huge fins and overstated design, was available in 1957 for approximately $1800. The line graph in Figure 1.28 shows the average fuel efficiency, in miles per gallon, of new U.S. passenger cars for selected years from 1955 through 2018. Based on the averages shown by the graph, it’s unlikely that this classic 1957 Chevy got more than 15 miles per gallon.
You are probably familiar with the words used to describe the graph in Figure 1.28:
In this section, you will enhance your intuitive understanding of ways of describing graphs by viewing these descriptions from the perspective of functions.
Objective 1: Identify intervals on which a function increases, decreases, or is constant
Increasing and Decreasing Functions
Objective 1 Identify intervals on which a function increases, decreases, or is constant.
Watch Video
Too late for that flu shot now! It’s only 8 a.m. and you’re feeling lousy. Your temperature is . Fascinated by the way that algebra models the world (your author is projecting a bit here), you decide to construct graphs showing your body temperature as a function of the time of day. You decide to let x represent the number of hours after 8 a.m. and your temperature at time x.
At 8 a.m. your temperature is and you are not feeling well. However, your temperature starts to decrease. It reaches normal by 11 a.m. Feeling energized, you construct the graph shown on the right, indicating decreasing temperature for , or on the interval (0, 3).
Did creating that first graph drain you of your energy? Your temperature starts to rise after 11 a.m. By 1 p.m., 5 hours after 8 a.m., your temperature reaches . However, you keep plotting points on your graph. At the right, we can see that your temperature increases for , or on the interval (3, 5).
The graph of f is decreasing to the left of and increasing to the right of . Thus, your temperature 3 hours after 8 a.m. was at its lowest point. Your relative minimum temperature was .
By 3 p.m., your temperature is no worse than it was at 1 p.m.: It is still . (Of course, it’s no better either.) Your temperature remained the same, or constant, for , or on the interval (5, 7).
The time-temperature flu scenario illustrates that a function f is increasing when its graph rises from left to right, decreasing when its graph falls from left to right, and remains constant when it neither rises nor falls. Let’s now provide a more precise algebraic description for these intuitive concepts.
Increasing, Decreasing, and Constant Functions
A function is increasing on an open interval, I, if whenever for any and in the interval.
A function is decreasing on an open interval, I, if whenever for any and in the interval.
A function is constant on an open interval, I, if for any and in the interval.
The function is decreasing on the interval , increasing on the interval (0, 2), and decreasing on the interval .
Although the function’s equations are not given, the graph indicates that the function is defined in two pieces. The part of the graph to the left of the y-axis shows that the function is constant on the interval . The part to the right of the y-axis shows that the function is increasing on the interval .
Objective 1: Identify intervals on which a function increases, decreases, or is constant
Increasing and Decreasing Functions
Objective 1 Identify intervals on which a function increases, decreases, or is constant.
Watch Video
Too late for that flu shot now! It’s only 8 a.m. and you’re feeling lousy. Your temperature is . Fascinated by the way that algebra models the world (your author is projecting a bit here), you decide to construct graphs showing your body temperature as a function of the time of day. You decide to let x represent the number of hours after 8 a.m. and your temperature at time x.
At 8 a.m. your temperature is and you are not feeling well. However, your temperature starts to decrease. It reaches normal by 11 a.m. Feeling energized, you construct the graph shown on the right, indicating decreasing temperature for , or on the interval (0, 3).
Did creating that first graph drain you of your energy? Your temperature starts to rise after 11 a.m. By 1 p.m., 5 hours after 8 a.m., your temperature reaches . However, you keep plotting points on your graph. At the right, we can see that your temperature increases for , or on the interval (3, 5).
The graph of f is decreasing to the left of and increasing to the right of . Thus, your temperature 3 hours after 8 a.m. was at its lowest point. Your relative minimum temperature was .
By 3 p.m., your temperature is no worse than it was at 1 p.m.: It is still . (Of course, it’s no better either.) Your temperature remained the same, or constant, for , or on the interval (5, 7).
The time-temperature flu scenario illustrates that a function f is increasing when its graph rises from left to right, decreasing when its graph falls from left to right, and remains constant when it neither rises nor falls. Let’s now provide a more precise algebraic description for these intuitive concepts.
Increasing, Decreasing, and Constant Functions
A function is increasing on an open interval, I, if whenever for any and in the interval.
A function is decreasing on an open interval, I, if whenever for any and in the interval.
A function is constant on an open interval, I, if for any and in the interval.
The function is decreasing on the interval , increasing on the interval (0, 2), and decreasing on the interval .
Although the function’s equations are not given, the graph indicates that the function is defined in two pieces. The part of the graph to the left of the y-axis shows that the function is constant on the interval . The part to the right of the y-axis shows that the function is increasing on the interval .
Objective 2: Use graphs to locate relative maxima or minima
Relative Maxima and Relative Minima
Objective 2 Use graphs to locate relative maxima or minima.
Watch Video
The points at which a function changes its increasing or decreasing behavior can be used to find any relative maximum or relative minimum values of the function.
Definitions of Relative Maximum and Relative Minimum
A function value is a relative maximum of f if there exists an open interval containing a such that for all in the open interval.
A function value is a relative minimum of f if there exists an open interval containing b such that for all in the open interval.
The word local is sometimes used instead of relative when describing maxima or minima.
If the graph of a function is given, we can often visually locate the number(s) at which the function has a relative maximum or a relative minimum. For example, the graph of f in Figure 1.29 shows that
f has a relative maximum at .
The relative maximum is .
f has a relative minimum at .
The relative minimum is .
Figure 1.29Using a graph to locate where a function has a relative maximum or minimum
Is beauty in the eye of the beholder? Or are there certain objects (or people) that are so well balanced and proportioned that they are universally pleasing to the eye? What constitutes an attractive human face? In Figure 1.30, we’ve drawn lines between paired features. Notice how the features line up almost perfectly. Each half of the face is a mirror image of the other half through the vertical line. If we superimpose a rectangular coordinate system on the attractive face, notice that a point on the right has a mirror image at on the left. The attractive face is said to be symmetric with respect to the y-axis.
Figure 1.30To most people, an attractive face is one in which each half is an almost perfect mirror image of the other half.
Did you know that graphs of some equations exhibit exactly the kind of symmetry shown by the attractive face in Figure 1.30, as well as other kinds of symmetry? The word symmetry comes from the Greek symmetria, meaning “the same measure.” Figure 1.31 shows three graphs, each with a common form of symmetry. Notice that the graph in Figure 1.31(a) shows the y-axis symmetry found in the attractive face.
Figure 1.31Graphs illustrating the three forms of symmetry
Table 1.2 defines three common forms of symmetry and gives rules to determine if the graph of an equation is symmetric with respect to the y-axis, the x-axis, or the origin.
Replacing x with and y with does not give the original equation. Thus, the graph of is not symmetric with respect to the origin.
Our symmetry tests reveal that the graph of is symmetric with respect to the x-axis only.
The graph of in Figure 1.32(a) shows the symmetry with respect to the x-axis that we determined in Example 2. Figure 1.32(b) illustrates that the graph fails the vertical line test. Consequently, y is not a function of x. We have seen that if a graph has x-axis symmetry, y is usually not a function of x.
Determine whether the graph of is symmetric with respect to the y-axis, the x-axis, or the origin.
Objective 3: Test for symmetry
Tests for Symmetry
Objective 3 Test for symmetry.
Watch Video
Is beauty in the eye of the beholder? Or are there certain objects (or people) that are so well balanced and proportioned that they are universally pleasing to the eye? What constitutes an attractive human face? In Figure 1.30, we’ve drawn lines between paired features. Notice how the features line up almost perfectly. Each half of the face is a mirror image of the other half through the vertical line. If we superimpose a rectangular coordinate system on the attractive face, notice that a point on the right has a mirror image at on the left. The attractive face is said to be symmetric with respect to the y-axis.
Figure 1.30To most people, an attractive face is one in which each half is an almost perfect mirror image of the other half.
Did you know that graphs of some equations exhibit exactly the kind of symmetry shown by the attractive face in Figure 1.30, as well as other kinds of symmetry? The word symmetry comes from the Greek symmetria, meaning “the same measure.” Figure 1.31 shows three graphs, each with a common form of symmetry. Notice that the graph in Figure 1.31(a) shows the y-axis symmetry found in the attractive face.
Figure 1.31Graphs illustrating the three forms of symmetry
Table 1.2 defines three common forms of symmetry and gives rules to determine if the graph of an equation is symmetric with respect to the y-axis, the x-axis, or the origin.
Replacing x with and y with does not give the original equation. Thus, the graph of is not symmetric with respect to the origin.
Our symmetry tests reveal that the graph of is symmetric with respect to the x-axis only.
The graph of in Figure 1.32(a) shows the symmetry with respect to the x-axis that we determined in Example 2. Figure 1.32(b) illustrates that the graph fails the vertical line test. Consequently, y is not a function of x. We have seen that if a graph has x-axis symmetry, y is usually not a function of x.
Determine whether the graph of is symmetric with respect to the y-axis, the x-axis, or the origin.
Objective 3: Test for symmetry
Tests for Symmetry
Objective 3 Test for symmetry.
Watch Video
Is beauty in the eye of the beholder? Or are there certain objects (or people) that are so well balanced and proportioned that they are universally pleasing to the eye? What constitutes an attractive human face? In Figure 1.30, we’ve drawn lines between paired features. Notice how the features line up almost perfectly. Each half of the face is a mirror image of the other half through the vertical line. If we superimpose a rectangular coordinate system on the attractive face, notice that a point on the right has a mirror image at on the left. The attractive face is said to be symmetric with respect to the y-axis.
Figure 1.30To most people, an attractive face is one in which each half is an almost perfect mirror image of the other half.
Did you know that graphs of some equations exhibit exactly the kind of symmetry shown by the attractive face in Figure 1.30, as well as other kinds of symmetry? The word symmetry comes from the Greek symmetria, meaning “the same measure.” Figure 1.31 shows three graphs, each with a common form of symmetry. Notice that the graph in Figure 1.31(a) shows the y-axis symmetry found in the attractive face.
Figure 1.31Graphs illustrating the three forms of symmetry
Table 1.2 defines three common forms of symmetry and gives rules to determine if the graph of an equation is symmetric with respect to the y-axis, the x-axis, or the origin.
Replacing x with and y with does not give the original equation. Thus, the graph of is not symmetric with respect to the origin.
Our symmetry tests reveal that the graph of is symmetric with respect to the x-axis only.
The graph of in Figure 1.32(a) shows the symmetry with respect to the x-axis that we determined in Example 2. Figure 1.32(b) illustrates that the graph fails the vertical line test. Consequently, y is not a function of x. We have seen that if a graph has x-axis symmetry, y is usually not a function of x.
Determine whether the graph of is symmetric with respect to the y-axis, the x-axis, or the origin.
Objective 4: Identify even or odd functions and recognize their symmetries
Even and Odd Functions
Objective 4 Identify even or odd functions and recognize their symmetries.
Watch Video
We have seen that if a graph is symmetric with respect to the x-axis, it usually fails the vertical line test and is not the graph of a function. However, many functions have graphs that are symmetric with respect to the y-axis or the origin. We give these functions special names.
A function whose graph is symmetric with respect to the y-axis is called an even function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an even function.
Even Functions and Their Symmetries
The function f is an even function if
The graph of an even function is symmetric with respect to the y-axis.
An example of an even function is . The graph, shown in Figure 1.35, is symmetric with respect to the y-axis.
Because and , we see that . This algebraically identifies the function as an even function.
A function whose graph is symmetric with respect to the origin is called an odd function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an odd function.
Odd Functions and Their Symmetries
The function f is an odd function if
The graph of an odd function is symmetric with respect to the origin.
An example of an odd function is . The graph, shown in Figure 1.36, is symmetric with respect to the origin.
Note that each graph passes the vertical line test and is therefore the graph of a function. In each case, use inspection to determine whether or not there is symmetry. If the graph is symmetric with respect to the y-axis, it is that of an even function. If the graph is symmetric with respect to the origin, it is that of an odd function.
In addition to inspecting graphs, we can also use a function’s equation and the definitions of even and odd to determine whether the function is even, odd, or neither.
Identifying Even or Odd Functions from Equations
Even function:
The right side of the equation of an even function does not change if x is replaced with .
Odd function:
Every term on the right side of the equation of an odd function changes sign if x is replaced with .
Neither even nor odd: and
The right side of the equation changes if x is replaced with , but not every term on the right side changes sign.
Example 5 Identifying Even or Odd Functions from Equations
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Solution
In each case, replace x with and simplify. If the right side of the equation stays the same, the function is even. If every term on the right side changes sign, the function is odd.
There are two terms on the right side of the given equation, , and each term changed sign when we replaced x with . Because , f is an odd function. The graph of f is symmetric with respect to the origin.
The right side of the equation of the given function, , did not change when we replaced x with . Because is an even function. The graph of g is symmetric with respect to the y-axis.
We use the given function’s equation, , to find .
The right side of the equation of the given function, , changed when we replaced x with . Thus, , so h is not an even function. The sign of each of the three terms in the equation for did not change when we replaced x with . Only the second term changed signs. Thus, , so h is not an odd function. We conclude that h is neither an even nor an odd function. The graph of h is neither symmetric with respect to the y-axis nor with respect to the origin.
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Objective 4: Identify even or odd functions and recognize their symmetries
Even and Odd Functions
Objective 4 Identify even or odd functions and recognize their symmetries.
Watch Video
We have seen that if a graph is symmetric with respect to the x-axis, it usually fails the vertical line test and is not the graph of a function. However, many functions have graphs that are symmetric with respect to the y-axis or the origin. We give these functions special names.
A function whose graph is symmetric with respect to the y-axis is called an even function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an even function.
Even Functions and Their Symmetries
The function f is an even function if
The graph of an even function is symmetric with respect to the y-axis.
An example of an even function is . The graph, shown in Figure 1.35, is symmetric with respect to the y-axis.
Because and , we see that . This algebraically identifies the function as an even function.
A function whose graph is symmetric with respect to the origin is called an odd function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an odd function.
Odd Functions and Their Symmetries
The function f is an odd function if
The graph of an odd function is symmetric with respect to the origin.
An example of an odd function is . The graph, shown in Figure 1.36, is symmetric with respect to the origin.
Note that each graph passes the vertical line test and is therefore the graph of a function. In each case, use inspection to determine whether or not there is symmetry. If the graph is symmetric with respect to the y-axis, it is that of an even function. If the graph is symmetric with respect to the origin, it is that of an odd function.
In addition to inspecting graphs, we can also use a function’s equation and the definitions of even and odd to determine whether the function is even, odd, or neither.
Identifying Even or Odd Functions from Equations
Even function:
The right side of the equation of an even function does not change if x is replaced with .
Odd function:
Every term on the right side of the equation of an odd function changes sign if x is replaced with .
Neither even nor odd: and
The right side of the equation changes if x is replaced with , but not every term on the right side changes sign.
Example 5 Identifying Even or Odd Functions from Equations
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Solution
In each case, replace x with and simplify. If the right side of the equation stays the same, the function is even. If every term on the right side changes sign, the function is odd.
There are two terms on the right side of the given equation, , and each term changed sign when we replaced x with . Because , f is an odd function. The graph of f is symmetric with respect to the origin.
The right side of the equation of the given function, , did not change when we replaced x with . Because is an even function. The graph of g is symmetric with respect to the y-axis.
We use the given function’s equation, , to find .
The right side of the equation of the given function, , changed when we replaced x with . Thus, , so h is not an even function. The sign of each of the three terms in the equation for did not change when we replaced x with . Only the second term changed signs. Thus, , so h is not an odd function. We conclude that h is neither an even nor an odd function. The graph of h is neither symmetric with respect to the y-axis nor with respect to the origin.
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Objective 4: Identify even or odd functions and recognize their symmetries
Even and Odd Functions
Objective 4 Identify even or odd functions and recognize their symmetries.
Watch Video
We have seen that if a graph is symmetric with respect to the x-axis, it usually fails the vertical line test and is not the graph of a function. However, many functions have graphs that are symmetric with respect to the y-axis or the origin. We give these functions special names.
A function whose graph is symmetric with respect to the y-axis is called an even function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an even function.
Even Functions and Their Symmetries
The function f is an even function if
The graph of an even function is symmetric with respect to the y-axis.
An example of an even function is . The graph, shown in Figure 1.35, is symmetric with respect to the y-axis.
Because and , we see that . This algebraically identifies the function as an even function.
A function whose graph is symmetric with respect to the origin is called an odd function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an odd function.
Odd Functions and Their Symmetries
The function f is an odd function if
The graph of an odd function is symmetric with respect to the origin.
An example of an odd function is . The graph, shown in Figure 1.36, is symmetric with respect to the origin.
Note that each graph passes the vertical line test and is therefore the graph of a function. In each case, use inspection to determine whether or not there is symmetry. If the graph is symmetric with respect to the y-axis, it is that of an even function. If the graph is symmetric with respect to the origin, it is that of an odd function.
In addition to inspecting graphs, we can also use a function’s equation and the definitions of even and odd to determine whether the function is even, odd, or neither.
Identifying Even or Odd Functions from Equations
Even function:
The right side of the equation of an even function does not change if x is replaced with .
Odd function:
Every term on the right side of the equation of an odd function changes sign if x is replaced with .
Neither even nor odd: and
The right side of the equation changes if x is replaced with , but not every term on the right side changes sign.
Example 5 Identifying Even or Odd Functions from Equations
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Solution
In each case, replace x with and simplify. If the right side of the equation stays the same, the function is even. If every term on the right side changes sign, the function is odd.
There are two terms on the right side of the given equation, , and each term changed sign when we replaced x with . Because , f is an odd function. The graph of f is symmetric with respect to the origin.
The right side of the equation of the given function, , did not change when we replaced x with . Because is an even function. The graph of g is symmetric with respect to the y-axis.
We use the given function’s equation, , to find .
The right side of the equation of the given function, , changed when we replaced x with . Thus, , so h is not an even function. The sign of each of the three terms in the equation for did not change when we replaced x with . Only the second term changed signs. Thus, , so h is not an odd function. We conclude that h is neither an even nor an odd function. The graph of h is neither symmetric with respect to the y-axis nor with respect to the origin.
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Objective 4: Identify even or odd functions and recognize their symmetries
Even and Odd Functions
Objective 4 Identify even or odd functions and recognize their symmetries.
Watch Video
We have seen that if a graph is symmetric with respect to the x-axis, it usually fails the vertical line test and is not the graph of a function. However, many functions have graphs that are symmetric with respect to the y-axis or the origin. We give these functions special names.
A function whose graph is symmetric with respect to the y-axis is called an even function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an even function.
Even Functions and Their Symmetries
The function f is an even function if
The graph of an even function is symmetric with respect to the y-axis.
An example of an even function is . The graph, shown in Figure 1.35, is symmetric with respect to the y-axis.
Because and , we see that . This algebraically identifies the function as an even function.
A function whose graph is symmetric with respect to the origin is called an odd function. If the point is on the graph of the function, then the point is also on the graph. Expressing this symmetry in function notation provides the definition of an odd function.
Odd Functions and Their Symmetries
The function f is an odd function if
The graph of an odd function is symmetric with respect to the origin.
An example of an odd function is . The graph, shown in Figure 1.36, is symmetric with respect to the origin.
Note that each graph passes the vertical line test and is therefore the graph of a function. In each case, use inspection to determine whether or not there is symmetry. If the graph is symmetric with respect to the y-axis, it is that of an even function. If the graph is symmetric with respect to the origin, it is that of an odd function.
In addition to inspecting graphs, we can also use a function’s equation and the definitions of even and odd to determine whether the function is even, odd, or neither.
Identifying Even or Odd Functions from Equations
Even function:
The right side of the equation of an even function does not change if x is replaced with .
Odd function:
Every term on the right side of the equation of an odd function changes sign if x is replaced with .
Neither even nor odd: and
The right side of the equation changes if x is replaced with , but not every term on the right side changes sign.
Example 5 Identifying Even or Odd Functions from Equations
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Solution
In each case, replace x with and simplify. If the right side of the equation stays the same, the function is even. If every term on the right side changes sign, the function is odd.
There are two terms on the right side of the given equation, , and each term changed sign when we replaced x with . Because , f is an odd function. The graph of f is symmetric with respect to the origin.
The right side of the equation of the given function, , did not change when we replaced x with . Because is an even function. The graph of g is symmetric with respect to the y-axis.
We use the given function’s equation, , to find .
The right side of the equation of the given function, , changed when we replaced x with . Thus, , so h is not an even function. The sign of each of the three terms in the equation for did not change when we replaced x with . Only the second term changed signs. Thus, , so h is not an odd function. We conclude that h is neither an even nor an odd function. The graph of h is neither symmetric with respect to the y-axis nor with respect to the origin.
Determine whether each of the following functions is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
Objective 5: Understand and use piecewise functions
Piecewise Functions
Objective 5 Understand and use piecewise functions.
Watch Video
A cellphone company offers the following plan for a mobile hotspot:
$40 per month includes 15 GB of data.
Additional data costs $0.60 per GB.
We can represent this plan mathematically by writing the total monthly cost, C, as a function of the amount of data used, g.
A function that is defined by two (or more) equations over a specified domain is called a piecewise function. Many Internet plans can be represented with piecewise functions. The graph of the piecewise function described above is shown in Figure 1.37.
To find , we let . Because 10 lies between 0 and 15, we use the first line of the piecewise function.
This means that with 10 GB of data, the monthly cost is $40. This can be visually represented by the point (10, 40) on the first piece of the graph in Figure 1.37.
To find , we let . Because 45 is greater than 15, we use the second line of the piecewise function.
Thus, . This means that with 45 GB of data, the monthly cost is $58. This can be visually represented by the point (45, 58) on the second piece of the graph in Figure 1.37.
Use the function in Example 6 to find and interpret each of the following:
.
Identify your solutions by points on the graph in Figure 1.37.
Example 7 Graphing a Piecewise Function
Graph the piecewise function defined by
Solution
We graph f in two parts, using a partial table of coordinates to create each piece. The tables of coordinates and the completed graph are shown in Figure 1.38.
We can use the graph of the piecewise function in Figure 1.38 to find the range of f. The range of the blue piece on the left is . The range of the red horizontal piece on the right is . Thus, the range of f is
Some piecewise functions are called step functions because their graphs form discontinuous steps. One such function is called the greatest integer function, symbolized by int(x) or , where
For example,
Here are some additional examples:
Notice how we jumped from 1 to 2 in the function values for int(x). In particular,
The graph of is shown in Figure 1.39. The graph of the greatest integer function jumps vertically one unit at each integer. However, the graph is constant between each pair of consecutive integers. The rightmost horizontal step shown in the graph illustrates that
Figure 1.39The graph of the greatest integer function
A function that is defined by two (or more) equations over a specified domain is called a piecewise function. Many Internet plans can be represented with piecewise functions. The graph of the piecewise function described above is shown in Figure 1.37.
To find , we let . Because 10 lies between 0 and 15, we use the first line of the piecewise function.
This means that with 10 GB of data, the monthly cost is $40. This can be visually represented by the point (10, 40) on the first piece of the graph in Figure 1.37.
To find , we let . Because 45 is greater than 15, we use the second line of the piecewise function.
Thus, . This means that with 45 GB of data, the monthly cost is $58. This can be visually represented by the point (45, 58) on the second piece of the graph in Figure 1.37.
Use the function in Example 6 to find and interpret each of the following:
.
Identify your solutions by points on the graph in Figure 1.37.
Example 7 Graphing a Piecewise Function
Graph the piecewise function defined by
Solution
We graph f in two parts, using a partial table of coordinates to create each piece. The tables of coordinates and the completed graph are shown in Figure 1.38.
We can use the graph of the piecewise function in Figure 1.38 to find the range of f. The range of the blue piece on the left is . The range of the red horizontal piece on the right is . Thus, the range of f is
Some piecewise functions are called step functions because their graphs form discontinuous steps. One such function is called the greatest integer function, symbolized by int(x) or , where
For example,
Here are some additional examples:
Notice how we jumped from 1 to 2 in the function values for int(x). In particular,
The graph of is shown in Figure 1.39. The graph of the greatest integer function jumps vertically one unit at each integer. However, the graph is constant between each pair of consecutive integers. The rightmost horizontal step shown in the graph illustrates that
Figure 1.39The graph of the greatest integer function
Objective 6: Find and simplify a function’s difference quotient
Functions and Difference Quotients
Objective 6 Find and simplify a function’s difference quotient.
Watch Video
In Section 1.5, we will be studying the average rate of change of a function. A ratio, called the difference quotient, plays an important role in understanding the rate at which functions change.
Definition of the Difference Quotient of a Function
The expression
for is called the difference quotient of the function f.
Example 8 Evaluating and Simplifying a Difference Quotient
If , find and simplify each expression:
.
Solution
We find by replacing x with each time that x appears in the equation.
In Exercises 17–32, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–36, use possible symmetry to determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
33.
34.
35.
36.
In Exercises 37–48, determine whether each function is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
the number at which f has a relative minimum
the relative minimum of f
the values of x for which
Is f even, odd, or neither?
50. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
the numbers at which f has a relative maximum
the relative maxima of f
the values of x for which
Is f even, odd, or neither?
51. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
any relative maxima and the numbers at which they occur
the value of x for which
Is positive or negative?
52. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
values of x for which
values of x for which
Is positive or negative?
Is f even, odd, or neither?
Is a relative maximum?
In Exercises 53–58, evaluate each piecewise function at the given values of the independent variable.
53.
54.
55.
56.
57.
58.
In Exercises 59–70, the domain of each piecewise function is .
Graph each function.
Use your graph to determine the function’s range.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
In Exercises 71–92, find and simplify the difference quotient
for the given function.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–94, let f be defined by the following graph:
A cable company offers the following high-speed Internet plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises 95–96.
Plan A
$40 per month includes 400 GB of data.
Additional data costs $0.20 per GB.
Plan B
$60 per month includes 1000 GB of data.
Additional data costs $0.20 per GB.
95. Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.
96. Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.
In Exercises 97–98, write a piecewise function that models each high-speed Internet plan. Then graph the function.
97. $50 per month includes 600 GB. Additional data costs $0.30 per GB.
98. $80 per month includes 2000 GB. Additional data costs $0.35 per GB.
Application Exercises
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.
Source: Thompson et al., The Science of Nutrition, Benjamin Cummings, 2008
99. State the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
100. State the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
101. For what age does the percent body fat in women reach a maximum? What is the percent body fat for that age?
102. At what age does the percent body fat in men reach a maximum? What is the percent body fat for that age?
103. Use interval notation to give the domain and the range for the graph of the function for women.
104. Use interval notation to give the domain and the range for the graph of the function for men.
105. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
106. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for 2020.
If Your Taxable Income Is Over
But Not Over
The Tax You Owe Is
Of the Amount Over
$ 0
$ 9,875
10%
$ 0
$ 9875
$ 40,125
$ 9875
$ 40,125
$ 85,525
$ 40,125
$ 85,525
$163,300
$ 85,525
$163,300
$207,350
$163,300
$207,350
$518,400
$207,350
$518,400
−
$518,400
The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and is the tax owed:
Use this information to solve Exercises 107–108.
107. Find and interpret .
108. Find and interpret .
In Exercises 109–110, refer to the preceding tax table.
109. Find the algebraic expression for the missing piece of that models tax owed for the domain .
110. Find the algebraic expression for the missing piece of that models tax owed for the domain .
The figure shows the cost of mailing a first-class letter, , as a function of its weight, x, in ounces, in June 2020. Use the graph to solve Exercises 111–114.
111. Find . What does this mean in terms of the variables in this situation?
112. Find . What does this mean in terms of the variables in this situation?
113. What is the cost of mailing a letter that weighs 1.5 ounces?
114. What is the cost of mailing a letter that weighs 1.8 ounces?
115. Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed 4, the monthly cost is $20. The cost then increases by $2 for each successive year of the pet’s age.
Age Not Exceeding
Monthly Cost
4
$20
5
$22
6
$24
The cost schedule continues in this manner for ages not exceeding 10. The cost for pets whose ages exceed 10 is $40. Use this information to create a graph that shows the monthly cost of the insurance, , for a pet of age x, where the function’s domain is .
Explaining the Concepts
116. What does it mean if a function f is increasing on an interval?
117. Suppose that a function f whose graph contains no breaks or gaps on is increasing on , decreasing on , and defined at b. Describe what occurs at . What does the function value represent?
118. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the y-axis?
119. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the x-axis?
120. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the origin?
121. If you are given a function’s graph, how do you determine if the function is even, odd, or neither?
122. If you are given a function’s equation, how do you determine if the function is even, odd, or neither?
123. What is a piecewise function?
124. Explain how to find the difference quotient of a function , if an equation for f is given.
Technology Exercises
125. The function
models the number of annual physician visits, , by a person of age x. Graph the function in a by viewing rectangle. What does the shape of the graph indicate about the relationship between one’s age and the number of annual physician visits? Use the or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
In Exercises 126–131, use a graphing utility to graph each function. Use a by viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant.
126.
127.
128.
129.
130.
131.
132.
Graph the functions for , and 6 in a by viewing rectangle.
Graph the functions for , and 5 in a by viewing rectangle.
If n is positive and even, where is the graph of increasing and where is it decreasing?
If n is positive and odd, what can you conclude about the graph of in terms of increasing or decreasing behavior?
Graph all six functions in a by viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?
Critical Thinking Exercises
Make Sense? In Exercises 133–136, determine whether each statement makes sense or does not make sense, and explain your reasoning.
133. My graph is decreasing on and increasing on , so must be a relative maximum.
134. This work by artist Scott Kim has the same kind of symmetry as an even function.
135. I graphed
and one piece of my graph is a single point.
136. I noticed that the difference quotient is always zero if , where c is any constant.
137. Sketch the graph of f using the following properties. (More than one correct graph is possible.) f is a piecewise function that is decreasing on is increasing on , and the range of f is .
138. Define a piecewise function on the intervals , (2, 5), and that does not “jump” at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.
139. Suppose that . The function f can be even, odd, or neither. The same is true for the function g.
Under what conditions is h definitely an even function?
Under what conditions is h definitely an odd function?
Retaining the Concepts
140. You invested $80,000 in two accounts paying 1.5% and 1.7% annual interest. If the total interest earned for the year was $1320, how much was invested at each rate? (Section P.8, Example 5)
In Exercises 17–32, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–36, use possible symmetry to determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
33.
34.
35.
36.
In Exercises 37–48, determine whether each function is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
the number at which f has a relative minimum
the relative minimum of f
the values of x for which
Is f even, odd, or neither?
50. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
the numbers at which f has a relative maximum
the relative maxima of f
the values of x for which
Is f even, odd, or neither?
51. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
any relative maxima and the numbers at which they occur
the value of x for which
Is positive or negative?
52. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
values of x for which
values of x for which
Is positive or negative?
Is f even, odd, or neither?
Is a relative maximum?
In Exercises 53–58, evaluate each piecewise function at the given values of the independent variable.
53.
54.
55.
56.
57.
58.
In Exercises 59–70, the domain of each piecewise function is .
Graph each function.
Use your graph to determine the function’s range.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
In Exercises 71–92, find and simplify the difference quotient
for the given function.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–94, let f be defined by the following graph:
A cable company offers the following high-speed Internet plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises 95–96.
Plan A
$40 per month includes 400 GB of data.
Additional data costs $0.20 per GB.
Plan B
$60 per month includes 1000 GB of data.
Additional data costs $0.20 per GB.
95. Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.
96. Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.
In Exercises 97–98, write a piecewise function that models each high-speed Internet plan. Then graph the function.
97. $50 per month includes 600 GB. Additional data costs $0.30 per GB.
98. $80 per month includes 2000 GB. Additional data costs $0.35 per GB.
Application Exercises
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.
Source: Thompson et al., The Science of Nutrition, Benjamin Cummings, 2008
99. State the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
100. State the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
101. For what age does the percent body fat in women reach a maximum? What is the percent body fat for that age?
102. At what age does the percent body fat in men reach a maximum? What is the percent body fat for that age?
103. Use interval notation to give the domain and the range for the graph of the function for women.
104. Use interval notation to give the domain and the range for the graph of the function for men.
105. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
106. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for 2020.
If Your Taxable Income Is Over
But Not Over
The Tax You Owe Is
Of the Amount Over
$ 0
$ 9,875
10%
$ 0
$ 9875
$ 40,125
$ 9875
$ 40,125
$ 85,525
$ 40,125
$ 85,525
$163,300
$ 85,525
$163,300
$207,350
$163,300
$207,350
$518,400
$207,350
$518,400
−
$518,400
The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and is the tax owed:
Use this information to solve Exercises 107–108.
107. Find and interpret .
108. Find and interpret .
In Exercises 109–110, refer to the preceding tax table.
109. Find the algebraic expression for the missing piece of that models tax owed for the domain .
110. Find the algebraic expression for the missing piece of that models tax owed for the domain .
The figure shows the cost of mailing a first-class letter, , as a function of its weight, x, in ounces, in June 2020. Use the graph to solve Exercises 111–114.
111. Find . What does this mean in terms of the variables in this situation?
112. Find . What does this mean in terms of the variables in this situation?
113. What is the cost of mailing a letter that weighs 1.5 ounces?
114. What is the cost of mailing a letter that weighs 1.8 ounces?
115. Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed 4, the monthly cost is $20. The cost then increases by $2 for each successive year of the pet’s age.
Age Not Exceeding
Monthly Cost
4
$20
5
$22
6
$24
The cost schedule continues in this manner for ages not exceeding 10. The cost for pets whose ages exceed 10 is $40. Use this information to create a graph that shows the monthly cost of the insurance, , for a pet of age x, where the function’s domain is .
Explaining the Concepts
116. What does it mean if a function f is increasing on an interval?
117. Suppose that a function f whose graph contains no breaks or gaps on is increasing on , decreasing on , and defined at b. Describe what occurs at . What does the function value represent?
118. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the y-axis?
119. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the x-axis?
120. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the origin?
121. If you are given a function’s graph, how do you determine if the function is even, odd, or neither?
122. If you are given a function’s equation, how do you determine if the function is even, odd, or neither?
123. What is a piecewise function?
124. Explain how to find the difference quotient of a function , if an equation for f is given.
Technology Exercises
125. The function
models the number of annual physician visits, , by a person of age x. Graph the function in a by viewing rectangle. What does the shape of the graph indicate about the relationship between one’s age and the number of annual physician visits? Use the or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
In Exercises 126–131, use a graphing utility to graph each function. Use a by viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant.
126.
127.
128.
129.
130.
131.
132.
Graph the functions for , and 6 in a by viewing rectangle.
Graph the functions for , and 5 in a by viewing rectangle.
If n is positive and even, where is the graph of increasing and where is it decreasing?
If n is positive and odd, what can you conclude about the graph of in terms of increasing or decreasing behavior?
Graph all six functions in a by viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?
Critical Thinking Exercises
Make Sense? In Exercises 133–136, determine whether each statement makes sense or does not make sense, and explain your reasoning.
133. My graph is decreasing on and increasing on , so must be a relative maximum.
134. This work by artist Scott Kim has the same kind of symmetry as an even function.
135. I graphed
and one piece of my graph is a single point.
136. I noticed that the difference quotient is always zero if , where c is any constant.
137. Sketch the graph of f using the following properties. (More than one correct graph is possible.) f is a piecewise function that is decreasing on is increasing on , and the range of f is .
138. Define a piecewise function on the intervals , (2, 5), and that does not “jump” at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.
139. Suppose that . The function f can be even, odd, or neither. The same is true for the function g.
Under what conditions is h definitely an even function?
Under what conditions is h definitely an odd function?
Retaining the Concepts
140. You invested $80,000 in two accounts paying 1.5% and 1.7% annual interest. If the total interest earned for the year was $1320, how much was invested at each rate? (Section P.8, Example 5)
In Exercises 17–32, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–36, use possible symmetry to determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
33.
34.
35.
36.
In Exercises 37–48, determine whether each function is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
the number at which f has a relative minimum
the relative minimum of f
the values of x for which
Is f even, odd, or neither?
50. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
the numbers at which f has a relative maximum
the relative maxima of f
the values of x for which
Is f even, odd, or neither?
51. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
any relative maxima and the numbers at which they occur
the value of x for which
Is positive or negative?
52. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
values of x for which
values of x for which
Is positive or negative?
Is f even, odd, or neither?
Is a relative maximum?
In Exercises 53–58, evaluate each piecewise function at the given values of the independent variable.
53.
54.
55.
56.
57.
58.
In Exercises 59–70, the domain of each piecewise function is .
Graph each function.
Use your graph to determine the function’s range.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
In Exercises 71–92, find and simplify the difference quotient
for the given function.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–94, let f be defined by the following graph:
A cable company offers the following high-speed Internet plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises 95–96.
Plan A
$40 per month includes 400 GB of data.
Additional data costs $0.20 per GB.
Plan B
$60 per month includes 1000 GB of data.
Additional data costs $0.20 per GB.
95. Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.
96. Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.
In Exercises 97–98, write a piecewise function that models each high-speed Internet plan. Then graph the function.
97. $50 per month includes 600 GB. Additional data costs $0.30 per GB.
98. $80 per month includes 2000 GB. Additional data costs $0.35 per GB.
Application Exercises
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.
Source: Thompson et al., The Science of Nutrition, Benjamin Cummings, 2008
99. State the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
100. State the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
101. For what age does the percent body fat in women reach a maximum? What is the percent body fat for that age?
102. At what age does the percent body fat in men reach a maximum? What is the percent body fat for that age?
103. Use interval notation to give the domain and the range for the graph of the function for women.
104. Use interval notation to give the domain and the range for the graph of the function for men.
105. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
106. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for 2020.
If Your Taxable Income Is Over
But Not Over
The Tax You Owe Is
Of the Amount Over
$ 0
$ 9,875
10%
$ 0
$ 9875
$ 40,125
$ 9875
$ 40,125
$ 85,525
$ 40,125
$ 85,525
$163,300
$ 85,525
$163,300
$207,350
$163,300
$207,350
$518,400
$207,350
$518,400
−
$518,400
The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and is the tax owed:
Use this information to solve Exercises 107–108.
107. Find and interpret .
108. Find and interpret .
In Exercises 109–110, refer to the preceding tax table.
109. Find the algebraic expression for the missing piece of that models tax owed for the domain .
110. Find the algebraic expression for the missing piece of that models tax owed for the domain .
The figure shows the cost of mailing a first-class letter, , as a function of its weight, x, in ounces, in June 2020. Use the graph to solve Exercises 111–114.
111. Find . What does this mean in terms of the variables in this situation?
112. Find . What does this mean in terms of the variables in this situation?
113. What is the cost of mailing a letter that weighs 1.5 ounces?
114. What is the cost of mailing a letter that weighs 1.8 ounces?
115. Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed 4, the monthly cost is $20. The cost then increases by $2 for each successive year of the pet’s age.
Age Not Exceeding
Monthly Cost
4
$20
5
$22
6
$24
The cost schedule continues in this manner for ages not exceeding 10. The cost for pets whose ages exceed 10 is $40. Use this information to create a graph that shows the monthly cost of the insurance, , for a pet of age x, where the function’s domain is .
Explaining the Concepts
116. What does it mean if a function f is increasing on an interval?
117. Suppose that a function f whose graph contains no breaks or gaps on is increasing on , decreasing on , and defined at b. Describe what occurs at . What does the function value represent?
118. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the y-axis?
119. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the x-axis?
120. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the origin?
121. If you are given a function’s graph, how do you determine if the function is even, odd, or neither?
122. If you are given a function’s equation, how do you determine if the function is even, odd, or neither?
123. What is a piecewise function?
124. Explain how to find the difference quotient of a function , if an equation for f is given.
Technology Exercises
125. The function
models the number of annual physician visits, , by a person of age x. Graph the function in a by viewing rectangle. What does the shape of the graph indicate about the relationship between one’s age and the number of annual physician visits? Use the or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
In Exercises 126–131, use a graphing utility to graph each function. Use a by viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant.
126.
127.
128.
129.
130.
131.
132.
Graph the functions for , and 6 in a by viewing rectangle.
Graph the functions for , and 5 in a by viewing rectangle.
If n is positive and even, where is the graph of increasing and where is it decreasing?
If n is positive and odd, what can you conclude about the graph of in terms of increasing or decreasing behavior?
Graph all six functions in a by viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?
Critical Thinking Exercises
Make Sense? In Exercises 133–136, determine whether each statement makes sense or does not make sense, and explain your reasoning.
133. My graph is decreasing on and increasing on , so must be a relative maximum.
134. This work by artist Scott Kim has the same kind of symmetry as an even function.
135. I graphed
and one piece of my graph is a single point.
136. I noticed that the difference quotient is always zero if , where c is any constant.
137. Sketch the graph of f using the following properties. (More than one correct graph is possible.) f is a piecewise function that is decreasing on is increasing on , and the range of f is .
138. Define a piecewise function on the intervals , (2, 5), and that does not “jump” at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.
139. Suppose that . The function f can be even, odd, or neither. The same is true for the function g.
Under what conditions is h definitely an even function?
Under what conditions is h definitely an odd function?
Retaining the Concepts
140. You invested $80,000 in two accounts paying 1.5% and 1.7% annual interest. If the total interest earned for the year was $1320, how much was invested at each rate? (Section P.8, Example 5)
In Exercises 17–32, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–36, use possible symmetry to determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
33.
34.
35.
36.
In Exercises 37–48, determine whether each function is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
the number at which f has a relative minimum
the relative minimum of f
the values of x for which
Is f even, odd, or neither?
50. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
the numbers at which f has a relative maximum
the relative maxima of f
the values of x for which
Is f even, odd, or neither?
51. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
any relative maxima and the numbers at which they occur
the value of x for which
Is positive or negative?
52. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
values of x for which
values of x for which
Is positive or negative?
Is f even, odd, or neither?
Is a relative maximum?
In Exercises 53–58, evaluate each piecewise function at the given values of the independent variable.
53.
54.
55.
56.
57.
58.
In Exercises 59–70, the domain of each piecewise function is .
Graph each function.
Use your graph to determine the function’s range.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
In Exercises 71–92, find and simplify the difference quotient
for the given function.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–94, let f be defined by the following graph:
A cable company offers the following high-speed Internet plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises 95–96.
Plan A
$40 per month includes 400 GB of data.
Additional data costs $0.20 per GB.
Plan B
$60 per month includes 1000 GB of data.
Additional data costs $0.20 per GB.
95. Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.
96. Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.
In Exercises 97–98, write a piecewise function that models each high-speed Internet plan. Then graph the function.
97. $50 per month includes 600 GB. Additional data costs $0.30 per GB.
98. $80 per month includes 2000 GB. Additional data costs $0.35 per GB.
Application Exercises
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.
Source: Thompson et al., The Science of Nutrition, Benjamin Cummings, 2008
99. State the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
100. State the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
101. For what age does the percent body fat in women reach a maximum? What is the percent body fat for that age?
102. At what age does the percent body fat in men reach a maximum? What is the percent body fat for that age?
103. Use interval notation to give the domain and the range for the graph of the function for women.
104. Use interval notation to give the domain and the range for the graph of the function for men.
105. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
106. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for 2020.
If Your Taxable Income Is Over
But Not Over
The Tax You Owe Is
Of the Amount Over
$ 0
$ 9,875
10%
$ 0
$ 9875
$ 40,125
$ 9875
$ 40,125
$ 85,525
$ 40,125
$ 85,525
$163,300
$ 85,525
$163,300
$207,350
$163,300
$207,350
$518,400
$207,350
$518,400
−
$518,400
The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and is the tax owed:
Use this information to solve Exercises 107–108.
107. Find and interpret .
108. Find and interpret .
In Exercises 109–110, refer to the preceding tax table.
109. Find the algebraic expression for the missing piece of that models tax owed for the domain .
110. Find the algebraic expression for the missing piece of that models tax owed for the domain .
The figure shows the cost of mailing a first-class letter, , as a function of its weight, x, in ounces, in June 2020. Use the graph to solve Exercises 111–114.
111. Find . What does this mean in terms of the variables in this situation?
112. Find . What does this mean in terms of the variables in this situation?
113. What is the cost of mailing a letter that weighs 1.5 ounces?
114. What is the cost of mailing a letter that weighs 1.8 ounces?
115. Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed 4, the monthly cost is $20. The cost then increases by $2 for each successive year of the pet’s age.
Age Not Exceeding
Monthly Cost
4
$20
5
$22
6
$24
The cost schedule continues in this manner for ages not exceeding 10. The cost for pets whose ages exceed 10 is $40. Use this information to create a graph that shows the monthly cost of the insurance, , for a pet of age x, where the function’s domain is .
Explaining the Concepts
116. What does it mean if a function f is increasing on an interval?
117. Suppose that a function f whose graph contains no breaks or gaps on is increasing on , decreasing on , and defined at b. Describe what occurs at . What does the function value represent?
118. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the y-axis?
119. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the x-axis?
120. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the origin?
121. If you are given a function’s graph, how do you determine if the function is even, odd, or neither?
122. If you are given a function’s equation, how do you determine if the function is even, odd, or neither?
123. What is a piecewise function?
124. Explain how to find the difference quotient of a function , if an equation for f is given.
Technology Exercises
125. The function
models the number of annual physician visits, , by a person of age x. Graph the function in a by viewing rectangle. What does the shape of the graph indicate about the relationship between one’s age and the number of annual physician visits? Use the or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
In Exercises 126–131, use a graphing utility to graph each function. Use a by viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant.
126.
127.
128.
129.
130.
131.
132.
Graph the functions for , and 6 in a by viewing rectangle.
Graph the functions for , and 5 in a by viewing rectangle.
If n is positive and even, where is the graph of increasing and where is it decreasing?
If n is positive and odd, what can you conclude about the graph of in terms of increasing or decreasing behavior?
Graph all six functions in a by viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?
Critical Thinking Exercises
Make Sense? In Exercises 133–136, determine whether each statement makes sense or does not make sense, and explain your reasoning.
133. My graph is decreasing on and increasing on , so must be a relative maximum.
134. This work by artist Scott Kim has the same kind of symmetry as an even function.
135. I graphed
and one piece of my graph is a single point.
136. I noticed that the difference quotient is always zero if , where c is any constant.
137. Sketch the graph of f using the following properties. (More than one correct graph is possible.) f is a piecewise function that is decreasing on is increasing on , and the range of f is .
138. Define a piecewise function on the intervals , (2, 5), and that does not “jump” at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.
139. Suppose that . The function f can be even, odd, or neither. The same is true for the function g.
Under what conditions is h definitely an even function?
Under what conditions is h definitely an odd function?
Retaining the Concepts
140. You invested $80,000 in two accounts paying 1.5% and 1.7% annual interest. If the total interest earned for the year was $1320, how much was invested at each rate? (Section P.8, Example 5)
In Exercises 17–32, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–36, use possible symmetry to determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
33.
34.
35.
36.
In Exercises 37–48, determine whether each function is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
the number at which f has a relative minimum
the relative minimum of f
the values of x for which
Is f even, odd, or neither?
50. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
the numbers at which f has a relative maximum
the relative maxima of f
the values of x for which
Is f even, odd, or neither?
51. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
any relative maxima and the numbers at which they occur
the value of x for which
Is positive or negative?
52. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
values of x for which
values of x for which
Is positive or negative?
Is f even, odd, or neither?
Is a relative maximum?
In Exercises 53–58, evaluate each piecewise function at the given values of the independent variable.
53.
54.
55.
56.
57.
58.
In Exercises 59–70, the domain of each piecewise function is .
Graph each function.
Use your graph to determine the function’s range.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
In Exercises 71–92, find and simplify the difference quotient
for the given function.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–94, let f be defined by the following graph:
A cable company offers the following high-speed Internet plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises 95–96.
Plan A
$40 per month includes 400 GB of data.
Additional data costs $0.20 per GB.
Plan B
$60 per month includes 1000 GB of data.
Additional data costs $0.20 per GB.
95. Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.
96. Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.
In Exercises 97–98, write a piecewise function that models each high-speed Internet plan. Then graph the function.
97. $50 per month includes 600 GB. Additional data costs $0.30 per GB.
98. $80 per month includes 2000 GB. Additional data costs $0.35 per GB.
Application Exercises
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.
Source: Thompson et al., The Science of Nutrition, Benjamin Cummings, 2008
99. State the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
100. State the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
101. For what age does the percent body fat in women reach a maximum? What is the percent body fat for that age?
102. At what age does the percent body fat in men reach a maximum? What is the percent body fat for that age?
103. Use interval notation to give the domain and the range for the graph of the function for women.
104. Use interval notation to give the domain and the range for the graph of the function for men.
105. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
106. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for 2020.
If Your Taxable Income Is Over
But Not Over
The Tax You Owe Is
Of the Amount Over
$ 0
$ 9,875
10%
$ 0
$ 9875
$ 40,125
$ 9875
$ 40,125
$ 85,525
$ 40,125
$ 85,525
$163,300
$ 85,525
$163,300
$207,350
$163,300
$207,350
$518,400
$207,350
$518,400
−
$518,400
The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and is the tax owed:
Use this information to solve Exercises 107–108.
107. Find and interpret .
108. Find and interpret .
In Exercises 109–110, refer to the preceding tax table.
109. Find the algebraic expression for the missing piece of that models tax owed for the domain .
110. Find the algebraic expression for the missing piece of that models tax owed for the domain .
The figure shows the cost of mailing a first-class letter, , as a function of its weight, x, in ounces, in June 2020. Use the graph to solve Exercises 111–114.
111. Find . What does this mean in terms of the variables in this situation?
112. Find . What does this mean in terms of the variables in this situation?
113. What is the cost of mailing a letter that weighs 1.5 ounces?
114. What is the cost of mailing a letter that weighs 1.8 ounces?
115. Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed 4, the monthly cost is $20. The cost then increases by $2 for each successive year of the pet’s age.
Age Not Exceeding
Monthly Cost
4
$20
5
$22
6
$24
The cost schedule continues in this manner for ages not exceeding 10. The cost for pets whose ages exceed 10 is $40. Use this information to create a graph that shows the monthly cost of the insurance, , for a pet of age x, where the function’s domain is .
Explaining the Concepts
116. What does it mean if a function f is increasing on an interval?
117. Suppose that a function f whose graph contains no breaks or gaps on is increasing on , decreasing on , and defined at b. Describe what occurs at . What does the function value represent?
118. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the y-axis?
119. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the x-axis?
120. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the origin?
121. If you are given a function’s graph, how do you determine if the function is even, odd, or neither?
122. If you are given a function’s equation, how do you determine if the function is even, odd, or neither?
123. What is a piecewise function?
124. Explain how to find the difference quotient of a function , if an equation for f is given.
Technology Exercises
125. The function
models the number of annual physician visits, , by a person of age x. Graph the function in a by viewing rectangle. What does the shape of the graph indicate about the relationship between one’s age and the number of annual physician visits? Use the or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
In Exercises 126–131, use a graphing utility to graph each function. Use a by viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant.
126.
127.
128.
129.
130.
131.
132.
Graph the functions for , and 6 in a by viewing rectangle.
Graph the functions for , and 5 in a by viewing rectangle.
If n is positive and even, where is the graph of increasing and where is it decreasing?
If n is positive and odd, what can you conclude about the graph of in terms of increasing or decreasing behavior?
Graph all six functions in a by viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?
Critical Thinking Exercises
Make Sense? In Exercises 133–136, determine whether each statement makes sense or does not make sense, and explain your reasoning.
133. My graph is decreasing on and increasing on , so must be a relative maximum.
134. This work by artist Scott Kim has the same kind of symmetry as an even function.
135. I graphed
and one piece of my graph is a single point.
136. I noticed that the difference quotient is always zero if , where c is any constant.
137. Sketch the graph of f using the following properties. (More than one correct graph is possible.) f is a piecewise function that is decreasing on is increasing on , and the range of f is .
138. Define a piecewise function on the intervals , (2, 5), and that does not “jump” at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.
139. Suppose that . The function f can be even, odd, or neither. The same is true for the function g.
Under what conditions is h definitely an even function?
Under what conditions is h definitely an odd function?
Retaining the Concepts
140. You invested $80,000 in two accounts paying 1.5% and 1.7% annual interest. If the total interest earned for the year was $1320, how much was invested at each rate? (Section P.8, Example 5)
In Exercises 17–32, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–36, use possible symmetry to determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
33.
34.
35.
36.
In Exercises 37–48, determine whether each function is even, odd, or neither. Then determine whether the function’s graph is symmetric with respect to the y-axis, the origin, or neither.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
the number at which f has a relative minimum
the relative minimum of f
the values of x for which
Is f even, odd, or neither?
50. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the x-intercepts
the y-intercept
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
the numbers at which f has a relative maximum
the relative maxima of f
the values of x for which
Is f even, odd, or neither?
51. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
values of x for which
any relative maxima and the numbers at which they occur
the value of x for which
Is positive or negative?
52. Use the graph of f to determine each of the following. Where applicable, use interval notation.
the domain of f
the range of f
the zeros of f
intervals on which f is increasing
intervals on which f is decreasing
intervals on which f is constant
values of x for which
values of x for which
Is positive or negative?
Is f even, odd, or neither?
Is a relative maximum?
In Exercises 53–58, evaluate each piecewise function at the given values of the independent variable.
53.
54.
55.
56.
57.
58.
In Exercises 59–70, the domain of each piecewise function is .
Graph each function.
Use your graph to determine the function’s range.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
In Exercises 71–92, find and simplify the difference quotient
for the given function.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
Practice PLUS
In Exercises 93–94, let f be defined by the following graph:
A cable company offers the following high-speed Internet plans. Also given are the piecewise functions that model these plans. Use this information to solve Exercises 95–96.
Plan A
$40 per month includes 400 GB of data.
Additional data costs $0.20 per GB.
Plan B
$60 per month includes 1000 GB of data.
Additional data costs $0.20 per GB.
95. Simplify the algebraic expression in the second line of the piecewise function for plan A. Then use point-plotting to graph the function.
96. Simplify the algebraic expression in the second line of the piecewise function for plan B. Then use point-plotting to graph the function.
In Exercises 97–98, write a piecewise function that models each high-speed Internet plan. Then graph the function.
97. $50 per month includes 600 GB. Additional data costs $0.30 per GB.
98. $80 per month includes 2000 GB. Additional data costs $0.35 per GB.
Application Exercises
With aging, body fat increases and muscle mass declines. The line graphs show the percent body fat in adult women and men as they age from 25 to 75 years. Use the graphs to solve Exercises 99–106.
Source: Thompson et al., The Science of Nutrition, Benjamin Cummings, 2008
99. State the intervals on which the graph giving the percent body fat in women is increasing and decreasing.
100. State the intervals on which the graph giving the percent body fat in men is increasing and decreasing.
101. For what age does the percent body fat in women reach a maximum? What is the percent body fat for that age?
102. At what age does the percent body fat in men reach a maximum? What is the percent body fat for that age?
103. Use interval notation to give the domain and the range for the graph of the function for women.
104. Use interval notation to give the domain and the range for the graph of the function for men.
105. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
106. The function models percent body fat, , where x is the number of years a person’s age exceeds 25. Use the graphs to determine whether this model describes percent body fat in women or in men.
Here is the Federal Tax Rate Schedule X that specifies the tax owed by a single taxpayer for 2020.
If Your Taxable Income Is Over
But Not Over
The Tax You Owe Is
Of the Amount Over
$ 0
$ 9,875
10%
$ 0
$ 9875
$ 40,125
$ 9875
$ 40,125
$ 85,525
$ 40,125
$ 85,525
$163,300
$ 85,525
$163,300
$207,350
$163,300
$207,350
$518,400
$207,350
$518,400
−
$518,400
The preceding tax table can be modeled by a piecewise function, where x represents the taxable income of a single taxpayer and is the tax owed:
Use this information to solve Exercises 107–108.
107. Find and interpret .
108. Find and interpret .
In Exercises 109–110, refer to the preceding tax table.
109. Find the algebraic expression for the missing piece of that models tax owed for the domain .
110. Find the algebraic expression for the missing piece of that models tax owed for the domain .
The figure shows the cost of mailing a first-class letter, , as a function of its weight, x, in ounces, in June 2020. Use the graph to solve Exercises 111–114.
111. Find . What does this mean in terms of the variables in this situation?
112. Find . What does this mean in terms of the variables in this situation?
113. What is the cost of mailing a letter that weighs 1.5 ounces?
114. What is the cost of mailing a letter that weighs 1.8 ounces?
115. Furry Finances. A pet insurance policy has a monthly rate that is a function of the age of the insured dog or cat. For pets whose age does not exceed 4, the monthly cost is $20. The cost then increases by $2 for each successive year of the pet’s age.
Age Not Exceeding
Monthly Cost
4
$20
5
$22
6
$24
The cost schedule continues in this manner for ages not exceeding 10. The cost for pets whose ages exceed 10 is $40. Use this information to create a graph that shows the monthly cost of the insurance, , for a pet of age x, where the function’s domain is .
Explaining the Concepts
116. What does it mean if a function f is increasing on an interval?
117. Suppose that a function f whose graph contains no breaks or gaps on is increasing on , decreasing on , and defined at b. Describe what occurs at . What does the function value represent?
118. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the y-axis?
119. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the x-axis?
120. Given an equation in x and y, how do you determine if its graph is symmetric with respect to the origin?
121. If you are given a function’s graph, how do you determine if the function is even, odd, or neither?
122. If you are given a function’s equation, how do you determine if the function is even, odd, or neither?
123. What is a piecewise function?
124. Explain how to find the difference quotient of a function , if an equation for f is given.
Technology Exercises
125. The function
models the number of annual physician visits, , by a person of age x. Graph the function in a by viewing rectangle. What does the shape of the graph indicate about the relationship between one’s age and the number of annual physician visits? Use the or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?
In Exercises 126–131, use a graphing utility to graph each function. Use a by viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant.
126.
127.
128.
129.
130.
131.
132.
Graph the functions for , and 6 in a by viewing rectangle.
Graph the functions for , and 5 in a by viewing rectangle.
If n is positive and even, where is the graph of increasing and where is it decreasing?
If n is positive and odd, what can you conclude about the graph of in terms of increasing or decreasing behavior?
Graph all six functions in a by viewing rectangle. What do you observe about the graphs in terms of how flat or how steep they are?
Critical Thinking Exercises
Make Sense? In Exercises 133–136, determine whether each statement makes sense or does not make sense, and explain your reasoning.
133. My graph is decreasing on and increasing on , so must be a relative maximum.
134. This work by artist Scott Kim has the same kind of symmetry as an even function.
135. I graphed
and one piece of my graph is a single point.
136. I noticed that the difference quotient is always zero if , where c is any constant.
137. Sketch the graph of f using the following properties. (More than one correct graph is possible.) f is a piecewise function that is decreasing on is increasing on , and the range of f is .
138. Define a piecewise function on the intervals , (2, 5), and that does not “jump” at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.
139. Suppose that . The function f can be even, odd, or neither. The same is true for the function g.
Under what conditions is h definitely an even function?
Under what conditions is h definitely an odd function?
Retaining the Concepts
140. You invested $80,000 in two accounts paying 1.5% and 1.7% annual interest. If the total interest earned for the year was $1320, how much was invested at each rate? (Section P.8, Example 5)
Is there a relationship between literacy and child mortality? As the percentage of adult females who are literate increases, does the mortality of children under five decrease? Figure 1.40 indicates that this is, indeed, the case. Each point in the figure represents one country.
Data presented in a visual form as a set of points is called a scatter plot. Also shown in Figure 1.40 is a line that passes through or near the points. A line that best fits the data points in a scatter plot is called a regression line. By writing the equation of this line, we can obtain a model for the data and make predictions about child mortality based on the percentage of literate adult females in a country.
Data often fall on or near a line. In this section, we will use functions to model such data and make predictions. We begin with a discussion of a line’s steepness.
Objective 1: Calculate a line’s slope
The Slope of a Line
Objective 1 Calculate a line’s slope.
Watch Video
Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the run) when moving from one fixed point to another along the line. To calculate the slope of a line, we use a ratio that compares the change in y (the rise) to the corresponding change in x (the run).
Definition of Slope
The slope of the line through the distinct points and is
It is common notation to let the letter m represent the slope of a line. The letter m is used because it is the first letter of the French verb monter, meaning “to rise” or “to ascend.”
Example 1 Using the Definition of Slope
Find the slope of the line passing through each pair of points:
and
and .
Solution
Let and We obtain the slope as follows:
The situation is illustrated in Figure 1.41. The slope of the line is 5. For every vertical change, or rise, of 5 units, there is a corresponding horizontal change, or run, of 1 unit. The slope is positive and the line rises from left to right.
To find the slope of the line passing through and , we can let and The slope of the line is computed as follows:
The situation is illustrated in Figure 1.42. The slope of the line is. For every vertical change of units (6 units down), there is a corresponding horizontal change of 5 units. The slope is negative and the line falls from left to right.
Find the slope of the line passing through each pair of points:
and
and .
Example 1 illustrates that a line with a positive slope is increasing and a line with a negative slope is decreasing. By contrast, a horizontal line is a constant function and has a slope of zero. A vertical line has no horizontal change, so in the formula for slope. Because we cannot divide by zero, the slope of a vertical line is undefined. This discussion is summarized in Table 1.3.
Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the run) when moving from one fixed point to another along the line. To calculate the slope of a line, we use a ratio that compares the change in y (the rise) to the corresponding change in x (the run).
Definition of Slope
The slope of the line through the distinct points and is
It is common notation to let the letter m represent the slope of a line. The letter m is used because it is the first letter of the French verb monter, meaning “to rise” or “to ascend.”
Example 1 Using the Definition of Slope
Find the slope of the line passing through each pair of points:
and
and .
Solution
Let and We obtain the slope as follows:
The situation is illustrated in Figure 1.41. The slope of the line is 5. For every vertical change, or rise, of 5 units, there is a corresponding horizontal change, or run, of 1 unit. The slope is positive and the line rises from left to right.
To find the slope of the line passing through and , we can let and The slope of the line is computed as follows:
The situation is illustrated in Figure 1.42. The slope of the line is. For every vertical change of units (6 units down), there is a corresponding horizontal change of 5 units. The slope is negative and the line falls from left to right.
Find the slope of the line passing through each pair of points:
and
and .
Example 1 illustrates that a line with a positive slope is increasing and a line with a negative slope is decreasing. By contrast, a horizontal line is a constant function and has a slope of zero. A vertical line has no horizontal change, so in the formula for slope. Because we cannot divide by zero, the slope of a vertical line is undefined. This discussion is summarized in Table 1.3.
Objective 2: Write the point-slope form of the equation of a line
The Point-Slope Form of the Equation of a Line
Objective 2 Write the point-slope form of the equation of a line.
Watch Video
We can use the slope of a line to obtain various forms of the line’s equation. For example, consider a nonvertical line that has slope m and that contains the point .
The line in Figure 1.43 has slope m and contains the point . Let represent any other point on the line.
Figure 1.43 A line passing through with slope m
Regardless of where the point is located, the steepness of the line in Figure 1.43 remains the same. Thus, the ratio for the slope stays a constant m. This means that for all points along the line
We can clear the fraction by multiplying both sides by , the least common denominator.
Now, if we reverse the two sides, we obtain the point-slope form of the equation of a line.
Point-Slope Form of the Equation of a Line
The point-slope form of the equation of a nonvertical line with slope m that passes through the point is
For example, the point-slope form of the equation of the line passing through (1, 5) with slope is
We will soon be expressing the equation of a nonvertical line in function notation. To do so, we need to solve the point-slope form of a line’s equation for y. Example 2 illustrates how to isolate y on one side of the equal sign.
Example 2 Writing an Equation for a Line in Point-Slope Form
Write an equation in point-slope form for the line with slope 4 that passes through the point . Then solve the equation for y.
Solution
We use the point-slope form of the equation of a line with , and .
Write an equation in point-slope form for the line with slope 6 that passes through the point . Then solve the equation for y.
Example 3 Writing an Equation for a Line in Point-Slope Form
Write an equation in point-slope form for the line passing through the points and . (See Figure 1.44.) Then solve the equation for y.
Figure 1.44 Write an equation in point-slope form for this line.
Solution
To use the point-slope form, we need to find the slope. The slope is the change in the y-coordinates divided by the corresponding change in the x-coordinates.
We can take either point on the line to be . Let’s use . Now, we are ready to write the point-slope form of the equation.
We now have an equation in point-slope form for the line shown in Figure 1.44. Now, we solve this equation for y.
Write an equation in point-slope form for the line passing through the points and . Then solve the equation for y.
Objective 2: Write the point-slope form of the equation of a line
The Point-Slope Form of the Equation of a Line
Objective 2 Write the point-slope form of the equation of a line.
Watch Video
We can use the slope of a line to obtain various forms of the line’s equation. For example, consider a nonvertical line that has slope m and that contains the point .
The line in Figure 1.43 has slope m and contains the point . Let represent any other point on the line.
Figure 1.43 A line passing through with slope m
Regardless of where the point is located, the steepness of the line in Figure 1.43 remains the same. Thus, the ratio for the slope stays a constant m. This means that for all points along the line
We can clear the fraction by multiplying both sides by , the least common denominator.
Now, if we reverse the two sides, we obtain the point-slope form of the equation of a line.
Point-Slope Form of the Equation of a Line
The point-slope form of the equation of a nonvertical line with slope m that passes through the point is
For example, the point-slope form of the equation of the line passing through (1, 5) with slope is
We will soon be expressing the equation of a nonvertical line in function notation. To do so, we need to solve the point-slope form of a line’s equation for y. Example 2 illustrates how to isolate y on one side of the equal sign.
Example 2 Writing an Equation for a Line in Point-Slope Form
Write an equation in point-slope form for the line with slope 4 that passes through the point . Then solve the equation for y.
Solution
We use the point-slope form of the equation of a line with , and .
Write an equation in point-slope form for the line with slope 6 that passes through the point . Then solve the equation for y.
Example 3 Writing an Equation for a Line in Point-Slope Form
Write an equation in point-slope form for the line passing through the points and . (See Figure 1.44.) Then solve the equation for y.
Figure 1.44 Write an equation in point-slope form for this line.
Solution
To use the point-slope form, we need to find the slope. The slope is the change in the y-coordinates divided by the corresponding change in the x-coordinates.
We can take either point on the line to be . Let’s use . Now, we are ready to write the point-slope form of the equation.
We now have an equation in point-slope form for the line shown in Figure 1.44. Now, we solve this equation for y.
Write an equation in point-slope form for the line passing through the points and . Then solve the equation for y.
Objective 3: Write and graph the slope-intercept form of the equation of a line
The Slope-Intercept Form of the Equation of a Line
Objective 3 Write and graph the slope-intercept form of the equation of a line.
Watch Video
Let’s write the point-slope form of the equation of a nonvertical line with slope m and y-intercept b. The line is shown in Figure 1.45. Because the y-intercept is b, the line passes through . We use the point-slope form with and .
Finally, we solve for y by adding b to both sides.
Thus, if a line’s equation is written as with y isolated on one side, the coefficient of x is the line’s slope and the constant term is the y-intercept. This form of a line’s equation is called the slope-intercept form of the line.
Slope-Intercept Form of the Equation of a Line
The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is
The slope-intercept form of a line’s equation, , can be expressed in function notation by replacing y with
We have seen that functions in this form are called linear functions. Thus, in the equation of a linear function, the coefficient of x is the line’s slope and the constant term is the y-intercept. Here are two examples:
If a linear function’s equation is in slope-intercept form, we can use the y-intercept and the slope to obtain its graph.
Graphing Using the Slope and y-Intercept
Plot the point containing the y-intercept on the y-axis. This is the point .
Obtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point.
Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.
Example 4 Graphing Using the Slope and y-Intercept
Graph the linear function: .
Solution
The equation of the line is in the form . We can find the slope, m, by identifying the coefficient of x. We can find the y-intercept, b, by identifying the constant term.
Now that we have identified the slope,, and the y-intercept, 2, we use the three-step procedure to graph the equation.
Step 1 PLOT THE POINT CONTAINING THE Y-INTERCEPT ON THE Y-AXIS. The y-intercept is 2. We plot , shown in Figure 1.46.
Figure 1.46 The graph of
Step 2 OBTAIN A SECOND POINT USING THE SLOPE, M. WRITE M AS A FRACTION, AND USE RISE OVER RUN, STARTING AT THE POINT CONTAINING THE Y-INTERCEPT, TO PLOT THIS POINT. The slope,, is already written as a fraction.
We plot the second point on the line by starting at , the first point. Based on the slope, we move 3 units down (the rise) and 2 units to the right (the run). This puts us at a second point on the line, , shown in Figure 1.46.
Step 3 USE A STRAIGHTEDGE TO DRAW A LINE THROUGH THE TWO POINTS. The graph of the linear function is shown as a blue line in Figure 1.46.
Objective 3: Write and graph the slope-intercept form of the equation of a line
The Slope-Intercept Form of the Equation of a Line
Objective 3 Write and graph the slope-intercept form of the equation of a line.
Watch Video
Let’s write the point-slope form of the equation of a nonvertical line with slope m and y-intercept b. The line is shown in Figure 1.45. Because the y-intercept is b, the line passes through . We use the point-slope form with and .
Finally, we solve for y by adding b to both sides.
Thus, if a line’s equation is written as with y isolated on one side, the coefficient of x is the line’s slope and the constant term is the y-intercept. This form of a line’s equation is called the slope-intercept form of the line.
Slope-Intercept Form of the Equation of a Line
The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is
The slope-intercept form of a line’s equation, , can be expressed in function notation by replacing y with
We have seen that functions in this form are called linear functions. Thus, in the equation of a linear function, the coefficient of x is the line’s slope and the constant term is the y-intercept. Here are two examples:
If a linear function’s equation is in slope-intercept form, we can use the y-intercept and the slope to obtain its graph.
Graphing Using the Slope and y-Intercept
Plot the point containing the y-intercept on the y-axis. This is the point .
Obtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point.
Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.
Example 4 Graphing Using the Slope and y-Intercept
Graph the linear function: .
Solution
The equation of the line is in the form . We can find the slope, m, by identifying the coefficient of x. We can find the y-intercept, b, by identifying the constant term.
Now that we have identified the slope,, and the y-intercept, 2, we use the three-step procedure to graph the equation.
Step 1 PLOT THE POINT CONTAINING THE Y-INTERCEPT ON THE Y-AXIS. The y-intercept is 2. We plot , shown in Figure 1.46.
Figure 1.46 The graph of
Step 2 OBTAIN A SECOND POINT USING THE SLOPE, M. WRITE M AS A FRACTION, AND USE RISE OVER RUN, STARTING AT THE POINT CONTAINING THE Y-INTERCEPT, TO PLOT THIS POINT. The slope,, is already written as a fraction.
We plot the second point on the line by starting at , the first point. Based on the slope, we move 3 units down (the rise) and 2 units to the right (the run). This puts us at a second point on the line, , shown in Figure 1.46.
Step 3 USE A STRAIGHTEDGE TO DRAW A LINE THROUGH THE TWO POINTS. The graph of the linear function is shown as a blue line in Figure 1.46.
If a line is horizontal, its slope is zero: . Thus, the equation becomes , where b is the y-intercept. All horizontal lines have equations of the form .
All ordered pairs that are solutions of have a value of y that is always . Any value can be used for x. In the table on the right, we have selected three of the possible values for , and 3. The table shows that three ordered pairs that are solutions of are , and . Drawing a line that passes through the three points gives the horizontal line shown in Figure 1.47.
A horizontal line is given by an equation of the form
where b is the y-intercept of the line. The slope of a horizontal line is zero.
Because any vertical line can intersect the graph of a horizontal line only once, a horizontal line is the graph of a function. Thus, we can express the equation as . This linear function is often called a constant function.
Next, let’s see what we can discover about the graph of an equation of the form by looking at an example.
All ordered pairs that are solutions of have a value of x that is always 2. Any value can be used for y. In the table on the right, we have selected three of the possible values for , and 3. The table shows that three ordered pairs that are solutions of are , and . Drawing a line that passes through the three points gives the vertical line shown in Figure 1.48.
Figure 1.48 The graph of
Does a vertical line represent the graph of a linear function? No. Look at the graph of in Figure 1.48. A vertical line drawn through intersects the graph infinitely many times. This shows that infinitely many outputs are associated with the input 2. No vertical line represents a linear function.
Equation of a Vertical Line
A vertical line is given by an equation of the form
where a is the x-intercept of the line. The slope of a vertical line is undefined.
Objective 5: Recognize and use the general form of a line’s equation
The General Form of the Equation of a Line
Objective 5 Recognize and use the general form of a line’s equation.
Watch Video
The vertical line whose equation is cannot be written in slope-intercept form, , because its slope is undefined. However, every line has an equation that can be expressed in the form . For example, can be expressed as , or . The equation is called the general form of the equation of a line.
General Form of the Equation of a Line
Every line has an equation that can be written in the general form
where , and C are real numbers, and A and B are not both zero.
If the equation of a nonvertical line is given in general form, it is possible to find the slope, m, and the y-intercept, b, for the line. We solve the equation for y, transforming it into the slope-intercept form . In this form, the coefficient of x is the slope of the line and the constant term is its y-intercept.
Example 7 Finding the Slope and the y-Intercept
Find the slope and the y-intercept of the line whose equation is .
Solution
The equation is given in general form. We begin by rewriting it in the form . We need to solve for y.
Find the slope and the y-intercept of the line whose equation is . Then use the y-intercept and the slope to graph the equation.
Objective 6: Use intercepts to graph the general form of a line’s equation
Using Intercepts to Graph Ax + By + C = 0
Objective 6 Use intercepts to graph the general form of a line’s equation.
Watch Video
Example 7 and Check Point 7 illustrate that one way to graph the general form of a line’s equation is to convert to slope-intercept form, . Then use the slope and the y-intercept to obtain the graph.
A second method for graphing uses intercepts. This method does not require rewriting the general form in a different form.
Using Intercepts to Graph
Find the x-intercept. Let and solve for x. Plot the point containing the x-intercept on the x-axis.
Find the y-intercept. Let and solve for y. Plot the point containing the y-intercept on the y-axis.
Use a straightedge to draw a line through the two points containing the intercepts. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions.
As long as none of A, B, and C is zero, the graph of will have distinct x- and y-intercepts, and this three-step method can be used to graph the equation.
Example 8 Using Intercepts to Graph a Linear Equation
Graph using intercepts: .
Solution
Step 1 FIND THE x-INTERCEPT. LET AND SOLVE FOR x.
The x-intercept is , so the line passes through or , as shown in Figure 1.49.
Figure 1.49 The graph of
Step 2 FIND THE y-INTERCEPT. LET AND SOLVE FOR y.
The y-intercept is , so the line passes through , as shown in Figure 1.49.
Step 3 GRAPH THE EQUATION BY DRAWING A LINE THROUGH THE TWO POINTS CONTAINING THE INTERCEPTS. The graph of is shown in Figure 1.49.
We’ve covered a lot of territory. Let’s take a moment to summarize the various forms for equations of lines.
Equations of Lines
1. Point-slope form
2. Slope-intercept form
or
3. Horizontal line
or
4. Vertical line
5. General form
Objective 7: Model data with linear functions and make predictions
Applications
Objective 7 Model data with linear functions and make predictions.
Watch Video
Linear functions are useful for modeling data that fall on or near a line.
Example 9 Modeling Global Warming
The amount of carbon dioxide in the atmosphere, measured in parts per million, has been increasing as a result of the burning of oil and coal. The buildup of gases and particles traps heat and raises the planet’s temperature. The bar graph in Figure 1.50(a) on the next page gives the average atmospheric concentration of carbon dioxide and the average global temperature for six selected years. The data are displayed as a set of six points in a rectangular coordinate system in Figure 1.50(b).
Figure 1.50(a)(b)
Source: National Oceanic and Atmospheric Administration
Shown on the scatter plot in Figure 1.50(b) is a line that passes through or near the six points. Write the slope-intercept form of this equation using function notation.
The preindustrial concentration of atmospheric carbon dioxide was 280 parts per million. The United Nations’ Intergovernmental Panel on Climate Change predicts global temperatures will rise between and if carbon dioxide concentration doubles from the preindustrial level. Compared to the average global temperature of for 2009, how well does the function from part (a) model this prediction?
Solution
The line in Figure 1.50(b) passes through and . We start by finding its slope.
The slope indicates that for each increase of one part per million in carbon dioxide concentration, the average global temperature is increasing by approximately .
Now we write the line’s equation in slope-intercept form.
A linear function that models average global temperature, , for an atmospheric carbon dioxide concentration of x parts per million is
If carbon dioxide concentration doubles from its preindustrial level of 280 parts per million, which many experts deem very likely, the concentration will reach , or 560 parts per million. We use the linear function to predict average global temperature at this concentration.
Our model projects an average global temperature of for a carbon dioxide concentration of 560 parts per million. Compared to the average global temperature of for 2009 shown in Figure 1.50(a) on the previous page, this is an increase of
This is consistent with a rise between and as predicted by the Intergovernmental Panel on Climate Change.
Use the data points and , shown, but not labeled, in Figure 1.50(b) to obtain a linear function that models average global temperature, , for an atmospheric carbon dioxide concentration of x parts per million. Round m to three decimal places and b to one decimal place. Then use the function to project average global temperature at a concentration of 600 parts per million.
Objective 7: Model data with linear functions and make predictions
Applications
Objective 7 Model data with linear functions and make predictions.
Watch Video
Linear functions are useful for modeling data that fall on or near a line.
Example 9 Modeling Global Warming
The amount of carbon dioxide in the atmosphere, measured in parts per million, has been increasing as a result of the burning of oil and coal. The buildup of gases and particles traps heat and raises the planet’s temperature. The bar graph in Figure 1.50(a) on the next page gives the average atmospheric concentration of carbon dioxide and the average global temperature for six selected years. The data are displayed as a set of six points in a rectangular coordinate system in Figure 1.50(b).
Figure 1.50(a)(b)
Source: National Oceanic and Atmospheric Administration
Shown on the scatter plot in Figure 1.50(b) is a line that passes through or near the six points. Write the slope-intercept form of this equation using function notation.
The preindustrial concentration of atmospheric carbon dioxide was 280 parts per million. The United Nations’ Intergovernmental Panel on Climate Change predicts global temperatures will rise between and if carbon dioxide concentration doubles from the preindustrial level. Compared to the average global temperature of for 2009, how well does the function from part (a) model this prediction?
Solution
The line in Figure 1.50(b) passes through and . We start by finding its slope.
The slope indicates that for each increase of one part per million in carbon dioxide concentration, the average global temperature is increasing by approximately .
Now we write the line’s equation in slope-intercept form.
A linear function that models average global temperature, , for an atmospheric carbon dioxide concentration of x parts per million is
If carbon dioxide concentration doubles from its preindustrial level of 280 parts per million, which many experts deem very likely, the concentration will reach , or 560 parts per million. We use the linear function to predict average global temperature at this concentration.
Our model projects an average global temperature of for a carbon dioxide concentration of 560 parts per million. Compared to the average global temperature of for 2009 shown in Figure 1.50(a) on the previous page, this is an increase of
This is consistent with a rise between and as predicted by the Intergovernmental Panel on Climate Change.
Use the data points and , shown, but not labeled, in Figure 1.50(b) to obtain a linear function that models average global temperature, , for an atmospheric carbon dioxide concentration of x parts per million. Round m to three decimal places and b to one decimal place. Then use the function to project average global temperature at a concentration of 600 parts per million.
1.4: Exercise Set
1.4 Exercise Set
Practice Exercises
In Exercises 1–10, find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
1. (4, 7) and (8, 10)
2. (2, 1) and (3, 4)
3. and (2, 2)
4. and (2, 4)
5. and
6. and
7. and
8. and
9. (5, 3) and
10. and (3, 5)
In Exercises 11–38, use the given conditions to write an equation for each line in point-slope form and slope-intercept form.
11. , passing through (3, 5)
12. , passing through (1, 3)
13. , passing through
14. , passing through
15. , passing through
16. , passing through
17. , passing through
18. , passing through
19. , passing through
20. , passing through
21. , passing through the origin
22. , passing through the origin
23. , passing through
24. , passing through
25. Passing through (1, 2) and (5, 10)
26. Passing through (3, 5) and (8, 15)
27. Passing through and (0, 3)
28. Passing through and (0, 2)
29. Passing through and (2, 4)
30. Passing through and
31. Passing through and (3, 6)
32. Passing through and
33. Passing through and
34. Passing through and
35. Passing through (2, 4) with
36. Passing through with
37. and
38. and
In Exercises 39–48, give the slope and y-intercept of each line whose equation is given. Then graph the linear function.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–58, graph each equation in a rectangular coordinate system.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
In Exercises 59–66,
Rewrite the given equation in slope-intercept form.
Give the slope and y-intercept.
Use the slope and y-intercept to graph the linear function.
59.
60.
61.
62.
63.
64.
65.
66.
In Exercises 67–72, use intercepts to graph each equation.
67.
68.
69.
70.
71.
72.
Practice PLUS
In Exercises 73–76, find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
73. and
74. and
75. and
76. and
In Exercises 77–78, give the slope and y-intercept of each line whose equation is given. Assume that .
77.
78.
In Exercises 79–80, find the value of y if the line through the two given points is to have the indicated slope.
79. and (1, 4),
80. and ,
In Exercises 81–82, graph each linear function.
81.
82.
83. If one point on a line is and the line’s slope is , find the y-intercept.
84. If one point on a line is and the line’s slope is , find the y-intercept.
Use the figure to make the lists in Exercises 85–86.
85. List the slopes , and in order of decreasing size.
86. List the y-intercepts , and in order of decreasing size.
Application Exercises
Americans’ trust in government and the media has generally been on a downward trend since pollsters first asked questions on these topics in the second half of the twentieth century. Trust in government hit an all-time low of 14% in 2014, while trust in the media bottomed out at 32% in 2016. The bar graph shows the percentage of Americans trusting in the government and the media for five selected years. The data are displayed as two sets of five points each, one scatter plot for the percentage of Americans trusting in the government and one for the percentage of Americans trusting in the media. Also shown for each scatter plot is a line that passes through or near the five points. Use these lines to solve Exercises 87–88.
87. In this exercise, you will use the red line for trust in government shown on the scatter plot to develop a model for the percentage of Americans trusting in government.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in government, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in government in 2025. Round to the nearest percent.
88. In this exercise, you will use the blue line for trust in media shown on the scatter plot to develop a model for the percentage of Americans trusting in media.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in media, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in media in 2025. Round to the nearest percent.
The bar graph gives the life expectancy for American men and women born in six selected years. In Exercises 89–90, you will use the data to obtain models for life expectancy and make predictions about how long American men and women will live in the future.
89. Use the data for males shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent male life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show male life expectancies for 1980 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American men born x years after 1960.
Use the function from part (b) to project the life expectancy of American men born in 2020.
90. Use the data for females shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent female life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show female life expectancies for 1970 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American women born x years after 1960. Round the slope to two decimal places.
Use the function from part (b) to project the life expectancy of American women born in 2020.
91. Shown, again, is the scatter plot that indicates a relationship between the percentage of adult females in a country who are literate and the mortality of children under five. Also shown is a line that passes through or near the points. Find a linear function that models the data by finding the slope-intercept form of the line’s equation. Use the function to make a prediction about child mortality based on the percentage of adult females in a country who are literate.
92. Just as money doesn’t buy happiness for individuals, the two don’t necessarily go together for countries either. However, the scatter plot does show a relationship between a country’s annual per capita income and the percentage of people in that country who call themselves “happy.”
Source: Richard Layard, Happiness: Lessons from a New Science, Penguin, 2005
Draw a line that fits the data so that the spread of the data points around the line is as small as possible. Use the coordinates of two points along your line to write the slope-intercept form of its equation. Express the equation in function notation and use the linear function to make a prediction about national happiness based on per capita income.
Explaining the Concepts
93. What is the slope of a line and how is it found?
94. Describe how to write the equation of a line if the coordinates of two points along the line are known.
95. Explain how to derive the slope-intercept form of a line’s equation, , from the point-slope form
96. Explain how to graph the equation . Can this equation be expressed in slope-intercept form? Explain.
97. Explain how to use the general form of a line’s equation to find the line’s slope and y-intercept.
98. Explain how to use intercepts to graph the general form of a line’s equation.
99. Take another look at the scatter plot in Exercise 91. Although there is a relationship between literacy and child mortality, we cannot conclude that increased literacy causes child mortality to decrease. Offer two or more possible explanations for the data in the scatter plot.
Technology Exercises
Use a graphing utility to graph each equation in Exercises 100–103. Then use the feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation.
100.
101.
102.
103.
104. Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 developed countries.
Use the statistical menu of your graphing utility to enter the 19 ordered pairs of data items shown in the table.
Use the scatter plot capability to draw a scatter plot of the data.
Select the linear regression option. Use your utility to obtain values for a and b for the equation of the regression line, . You may also be given a correlation coefficient, r. Values of r close to 1 indicate that the points can be described by a linear relationship and the regression line has a positive slope. Values of r close to indicate that the points can be described by a linear relationship and the regression line has a negative slope. Values of r close to 0 indicate no linear relationship between the variables. In this case, a linear model does not accurately describe the data.
Use the appropriate sequence (consult your manual) to graph the regression equation on top of the points in the scatter plot.
Critical Thinking Exercises
Make Sense? In Exercises 105–108, determine whether each statement makes sense or does not make sense, and explain your reasoning.
105. The graph of my linear function at first increased, reached a maximum point, and then decreased.
106. A linear function that models tuition and fees at public four-year colleges from 2000 through 2020 has negative slope.
107. Because the variable m does not appear in , equations in this form make it impossible to determine the line’s slope.
108. The federal minimum wage was $7.25 per hour from 2009 through 2020, so models the minimum wage, , in dollars, for the domain .
In Exercises 109–112, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
109. The equation shows that no line can have a y-intercept that is numerically equal to its slope.
110. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.
111. The graph of the linear function is a line passing through the point (6, 0) with slope.
112. The graph of in the rectangular coordinate system is the single point (7, 0).
In Exercises 113–114, find the coefficients that must be placed in each shaded area so that the function’s graph will be a line satisfying the specified conditions.
113.
114.
115. Prove that the equation of a line passing through and can be written in the form . Why is this called the intercept form of a line?
116. Excited about the success of celebrity stamps, post office officials were rumored to have put forth a plan to institute two new types of thermometers. On these new scales, represents degrees Usher and represents degrees Rihanna. If it is known that , , and degrees Usher is linearly related to degrees Rihanna, write an equation expressing U in terms of R.
Group Exercise
117. In Exercises 87–88, we used the data in a bar graph to develop linear functions that modeled trust in government and trust in media. For this group exercise, you might find it helpful to pattern your work after Exercises 87 and 88. Group members should begin by consulting an almanac, newspaper, magazine, or the Internet to find data that appear to lie approximately on or near a line. Working by hand or using a graphing utility, group members should construct scatter plots for the data that were assembled. If working by hand, draw a line that approximately fits the data in each scatter plot and then write its equation as a function in slope-intercept form. If using a graphing utility, obtain the equation of each regression line. Then use each linear function’s equation to make predictions about what might occur in the future. Are there circumstances that might affect the accuracy of the prediction? List some of these circumstances.
Retaining the Concepts
118. According to the U.S. Office of Management and Budget, the cost of maintaining existing public transportation infrastructure in 2020 was $103.4 billion and is projected to increase by $0.9 billion each year. By which year is the cost of maintaining existing public transportation infrastructure expected to reach $116 billion?
Exercises 121–123 will help you prepare for the material covered in the next section.
121. Write the slope-intercept form of the equation of the line passing through whose slope is the same as the line whose equation is .
122. Write an equation in general form of the line passing through whose slope is the negative reciprocal (the reciprocal with the opposite sign) of.
123. If , find
where and .
1.4: Exercise Set
1.4 Exercise Set
Practice Exercises
In Exercises 1–10, find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
1. (4, 7) and (8, 10)
2. (2, 1) and (3, 4)
3. and (2, 2)
4. and (2, 4)
5. and
6. and
7. and
8. and
9. (5, 3) and
10. and (3, 5)
In Exercises 11–38, use the given conditions to write an equation for each line in point-slope form and slope-intercept form.
11. , passing through (3, 5)
12. , passing through (1, 3)
13. , passing through
14. , passing through
15. , passing through
16. , passing through
17. , passing through
18. , passing through
19. , passing through
20. , passing through
21. , passing through the origin
22. , passing through the origin
23. , passing through
24. , passing through
25. Passing through (1, 2) and (5, 10)
26. Passing through (3, 5) and (8, 15)
27. Passing through and (0, 3)
28. Passing through and (0, 2)
29. Passing through and (2, 4)
30. Passing through and
31. Passing through and (3, 6)
32. Passing through and
33. Passing through and
34. Passing through and
35. Passing through (2, 4) with
36. Passing through with
37. and
38. and
In Exercises 39–48, give the slope and y-intercept of each line whose equation is given. Then graph the linear function.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–58, graph each equation in a rectangular coordinate system.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
In Exercises 59–66,
Rewrite the given equation in slope-intercept form.
Give the slope and y-intercept.
Use the slope and y-intercept to graph the linear function.
59.
60.
61.
62.
63.
64.
65.
66.
In Exercises 67–72, use intercepts to graph each equation.
67.
68.
69.
70.
71.
72.
Practice PLUS
In Exercises 73–76, find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
73. and
74. and
75. and
76. and
In Exercises 77–78, give the slope and y-intercept of each line whose equation is given. Assume that .
77.
78.
In Exercises 79–80, find the value of y if the line through the two given points is to have the indicated slope.
79. and (1, 4),
80. and ,
In Exercises 81–82, graph each linear function.
81.
82.
83. If one point on a line is and the line’s slope is , find the y-intercept.
84. If one point on a line is and the line’s slope is , find the y-intercept.
Use the figure to make the lists in Exercises 85–86.
85. List the slopes , and in order of decreasing size.
86. List the y-intercepts , and in order of decreasing size.
Application Exercises
Americans’ trust in government and the media has generally been on a downward trend since pollsters first asked questions on these topics in the second half of the twentieth century. Trust in government hit an all-time low of 14% in 2014, while trust in the media bottomed out at 32% in 2016. The bar graph shows the percentage of Americans trusting in the government and the media for five selected years. The data are displayed as two sets of five points each, one scatter plot for the percentage of Americans trusting in the government and one for the percentage of Americans trusting in the media. Also shown for each scatter plot is a line that passes through or near the five points. Use these lines to solve Exercises 87–88.
87. In this exercise, you will use the red line for trust in government shown on the scatter plot to develop a model for the percentage of Americans trusting in government.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in government, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in government in 2025. Round to the nearest percent.
88. In this exercise, you will use the blue line for trust in media shown on the scatter plot to develop a model for the percentage of Americans trusting in media.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in media, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in media in 2025. Round to the nearest percent.
The bar graph gives the life expectancy for American men and women born in six selected years. In Exercises 89–90, you will use the data to obtain models for life expectancy and make predictions about how long American men and women will live in the future.
89. Use the data for males shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent male life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show male life expectancies for 1980 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American men born x years after 1960.
Use the function from part (b) to project the life expectancy of American men born in 2020.
90. Use the data for females shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent female life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show female life expectancies for 1970 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American women born x years after 1960. Round the slope to two decimal places.
Use the function from part (b) to project the life expectancy of American women born in 2020.
91. Shown, again, is the scatter plot that indicates a relationship between the percentage of adult females in a country who are literate and the mortality of children under five. Also shown is a line that passes through or near the points. Find a linear function that models the data by finding the slope-intercept form of the line’s equation. Use the function to make a prediction about child mortality based on the percentage of adult females in a country who are literate.
92. Just as money doesn’t buy happiness for individuals, the two don’t necessarily go together for countries either. However, the scatter plot does show a relationship between a country’s annual per capita income and the percentage of people in that country who call themselves “happy.”
Source: Richard Layard, Happiness: Lessons from a New Science, Penguin, 2005
Draw a line that fits the data so that the spread of the data points around the line is as small as possible. Use the coordinates of two points along your line to write the slope-intercept form of its equation. Express the equation in function notation and use the linear function to make a prediction about national happiness based on per capita income.
Explaining the Concepts
93. What is the slope of a line and how is it found?
94. Describe how to write the equation of a line if the coordinates of two points along the line are known.
95. Explain how to derive the slope-intercept form of a line’s equation, , from the point-slope form
96. Explain how to graph the equation . Can this equation be expressed in slope-intercept form? Explain.
97. Explain how to use the general form of a line’s equation to find the line’s slope and y-intercept.
98. Explain how to use intercepts to graph the general form of a line’s equation.
99. Take another look at the scatter plot in Exercise 91. Although there is a relationship between literacy and child mortality, we cannot conclude that increased literacy causes child mortality to decrease. Offer two or more possible explanations for the data in the scatter plot.
Technology Exercises
Use a graphing utility to graph each equation in Exercises 100–103. Then use the feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation.
100.
101.
102.
103.
104. Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 developed countries.
Use the statistical menu of your graphing utility to enter the 19 ordered pairs of data items shown in the table.
Use the scatter plot capability to draw a scatter plot of the data.
Select the linear regression option. Use your utility to obtain values for a and b for the equation of the regression line, . You may also be given a correlation coefficient, r. Values of r close to 1 indicate that the points can be described by a linear relationship and the regression line has a positive slope. Values of r close to indicate that the points can be described by a linear relationship and the regression line has a negative slope. Values of r close to 0 indicate no linear relationship between the variables. In this case, a linear model does not accurately describe the data.
Use the appropriate sequence (consult your manual) to graph the regression equation on top of the points in the scatter plot.
Critical Thinking Exercises
Make Sense? In Exercises 105–108, determine whether each statement makes sense or does not make sense, and explain your reasoning.
105. The graph of my linear function at first increased, reached a maximum point, and then decreased.
106. A linear function that models tuition and fees at public four-year colleges from 2000 through 2020 has negative slope.
107. Because the variable m does not appear in , equations in this form make it impossible to determine the line’s slope.
108. The federal minimum wage was $7.25 per hour from 2009 through 2020, so models the minimum wage, , in dollars, for the domain .
In Exercises 109–112, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
109. The equation shows that no line can have a y-intercept that is numerically equal to its slope.
110. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.
111. The graph of the linear function is a line passing through the point (6, 0) with slope.
112. The graph of in the rectangular coordinate system is the single point (7, 0).
In Exercises 113–114, find the coefficients that must be placed in each shaded area so that the function’s graph will be a line satisfying the specified conditions.
113.
114.
115. Prove that the equation of a line passing through and can be written in the form . Why is this called the intercept form of a line?
116. Excited about the success of celebrity stamps, post office officials were rumored to have put forth a plan to institute two new types of thermometers. On these new scales, represents degrees Usher and represents degrees Rihanna. If it is known that , , and degrees Usher is linearly related to degrees Rihanna, write an equation expressing U in terms of R.
Group Exercise
117. In Exercises 87–88, we used the data in a bar graph to develop linear functions that modeled trust in government and trust in media. For this group exercise, you might find it helpful to pattern your work after Exercises 87 and 88. Group members should begin by consulting an almanac, newspaper, magazine, or the Internet to find data that appear to lie approximately on or near a line. Working by hand or using a graphing utility, group members should construct scatter plots for the data that were assembled. If working by hand, draw a line that approximately fits the data in each scatter plot and then write its equation as a function in slope-intercept form. If using a graphing utility, obtain the equation of each regression line. Then use each linear function’s equation to make predictions about what might occur in the future. Are there circumstances that might affect the accuracy of the prediction? List some of these circumstances.
Retaining the Concepts
118. According to the U.S. Office of Management and Budget, the cost of maintaining existing public transportation infrastructure in 2020 was $103.4 billion and is projected to increase by $0.9 billion each year. By which year is the cost of maintaining existing public transportation infrastructure expected to reach $116 billion?
Exercises 121–123 will help you prepare for the material covered in the next section.
121. Write the slope-intercept form of the equation of the line passing through whose slope is the same as the line whose equation is .
122. Write an equation in general form of the line passing through whose slope is the negative reciprocal (the reciprocal with the opposite sign) of.
123. If , find
where and .
1.4: Exercise Set
1.4 Exercise Set
Practice Exercises
In Exercises 1–10, find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
1. (4, 7) and (8, 10)
2. (2, 1) and (3, 4)
3. and (2, 2)
4. and (2, 4)
5. and
6. and
7. and
8. and
9. (5, 3) and
10. and (3, 5)
In Exercises 11–38, use the given conditions to write an equation for each line in point-slope form and slope-intercept form.
11. , passing through (3, 5)
12. , passing through (1, 3)
13. , passing through
14. , passing through
15. , passing through
16. , passing through
17. , passing through
18. , passing through
19. , passing through
20. , passing through
21. , passing through the origin
22. , passing through the origin
23. , passing through
24. , passing through
25. Passing through (1, 2) and (5, 10)
26. Passing through (3, 5) and (8, 15)
27. Passing through and (0, 3)
28. Passing through and (0, 2)
29. Passing through and (2, 4)
30. Passing through and
31. Passing through and (3, 6)
32. Passing through and
33. Passing through and
34. Passing through and
35. Passing through (2, 4) with
36. Passing through with
37. and
38. and
In Exercises 39–48, give the slope and y-intercept of each line whose equation is given. Then graph the linear function.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–58, graph each equation in a rectangular coordinate system.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
In Exercises 59–66,
Rewrite the given equation in slope-intercept form.
Give the slope and y-intercept.
Use the slope and y-intercept to graph the linear function.
59.
60.
61.
62.
63.
64.
65.
66.
In Exercises 67–72, use intercepts to graph each equation.
67.
68.
69.
70.
71.
72.
Practice PLUS
In Exercises 73–76, find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
73. and
74. and
75. and
76. and
In Exercises 77–78, give the slope and y-intercept of each line whose equation is given. Assume that .
77.
78.
In Exercises 79–80, find the value of y if the line through the two given points is to have the indicated slope.
79. and (1, 4),
80. and ,
In Exercises 81–82, graph each linear function.
81.
82.
83. If one point on a line is and the line’s slope is , find the y-intercept.
84. If one point on a line is and the line’s slope is , find the y-intercept.
Use the figure to make the lists in Exercises 85–86.
85. List the slopes , and in order of decreasing size.
86. List the y-intercepts , and in order of decreasing size.
Application Exercises
Americans’ trust in government and the media has generally been on a downward trend since pollsters first asked questions on these topics in the second half of the twentieth century. Trust in government hit an all-time low of 14% in 2014, while trust in the media bottomed out at 32% in 2016. The bar graph shows the percentage of Americans trusting in the government and the media for five selected years. The data are displayed as two sets of five points each, one scatter plot for the percentage of Americans trusting in the government and one for the percentage of Americans trusting in the media. Also shown for each scatter plot is a line that passes through or near the five points. Use these lines to solve Exercises 87–88.
87. In this exercise, you will use the red line for trust in government shown on the scatter plot to develop a model for the percentage of Americans trusting in government.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in government, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in government in 2025. Round to the nearest percent.
88. In this exercise, you will use the blue line for trust in media shown on the scatter plot to develop a model for the percentage of Americans trusting in media.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in media, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in media in 2025. Round to the nearest percent.
The bar graph gives the life expectancy for American men and women born in six selected years. In Exercises 89–90, you will use the data to obtain models for life expectancy and make predictions about how long American men and women will live in the future.
89. Use the data for males shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent male life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show male life expectancies for 1980 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American men born x years after 1960.
Use the function from part (b) to project the life expectancy of American men born in 2020.
90. Use the data for females shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent female life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show female life expectancies for 1970 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American women born x years after 1960. Round the slope to two decimal places.
Use the function from part (b) to project the life expectancy of American women born in 2020.
91. Shown, again, is the scatter plot that indicates a relationship between the percentage of adult females in a country who are literate and the mortality of children under five. Also shown is a line that passes through or near the points. Find a linear function that models the data by finding the slope-intercept form of the line’s equation. Use the function to make a prediction about child mortality based on the percentage of adult females in a country who are literate.
92. Just as money doesn’t buy happiness for individuals, the two don’t necessarily go together for countries either. However, the scatter plot does show a relationship between a country’s annual per capita income and the percentage of people in that country who call themselves “happy.”
Source: Richard Layard, Happiness: Lessons from a New Science, Penguin, 2005
Draw a line that fits the data so that the spread of the data points around the line is as small as possible. Use the coordinates of two points along your line to write the slope-intercept form of its equation. Express the equation in function notation and use the linear function to make a prediction about national happiness based on per capita income.
Explaining the Concepts
93. What is the slope of a line and how is it found?
94. Describe how to write the equation of a line if the coordinates of two points along the line are known.
95. Explain how to derive the slope-intercept form of a line’s equation, , from the point-slope form
96. Explain how to graph the equation . Can this equation be expressed in slope-intercept form? Explain.
97. Explain how to use the general form of a line’s equation to find the line’s slope and y-intercept.
98. Explain how to use intercepts to graph the general form of a line’s equation.
99. Take another look at the scatter plot in Exercise 91. Although there is a relationship between literacy and child mortality, we cannot conclude that increased literacy causes child mortality to decrease. Offer two or more possible explanations for the data in the scatter plot.
Technology Exercises
Use a graphing utility to graph each equation in Exercises 100–103. Then use the feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation.
100.
101.
102.
103.
104. Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 developed countries.
Use the statistical menu of your graphing utility to enter the 19 ordered pairs of data items shown in the table.
Use the scatter plot capability to draw a scatter plot of the data.
Select the linear regression option. Use your utility to obtain values for a and b for the equation of the regression line, . You may also be given a correlation coefficient, r. Values of r close to 1 indicate that the points can be described by a linear relationship and the regression line has a positive slope. Values of r close to indicate that the points can be described by a linear relationship and the regression line has a negative slope. Values of r close to 0 indicate no linear relationship between the variables. In this case, a linear model does not accurately describe the data.
Use the appropriate sequence (consult your manual) to graph the regression equation on top of the points in the scatter plot.
Critical Thinking Exercises
Make Sense? In Exercises 105–108, determine whether each statement makes sense or does not make sense, and explain your reasoning.
105. The graph of my linear function at first increased, reached a maximum point, and then decreased.
106. A linear function that models tuition and fees at public four-year colleges from 2000 through 2020 has negative slope.
107. Because the variable m does not appear in , equations in this form make it impossible to determine the line’s slope.
108. The federal minimum wage was $7.25 per hour from 2009 through 2020, so models the minimum wage, , in dollars, for the domain .
In Exercises 109–112, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
109. The equation shows that no line can have a y-intercept that is numerically equal to its slope.
110. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.
111. The graph of the linear function is a line passing through the point (6, 0) with slope.
112. The graph of in the rectangular coordinate system is the single point (7, 0).
In Exercises 113–114, find the coefficients that must be placed in each shaded area so that the function’s graph will be a line satisfying the specified conditions.
113.
114.
115. Prove that the equation of a line passing through and can be written in the form . Why is this called the intercept form of a line?
116. Excited about the success of celebrity stamps, post office officials were rumored to have put forth a plan to institute two new types of thermometers. On these new scales, represents degrees Usher and represents degrees Rihanna. If it is known that , , and degrees Usher is linearly related to degrees Rihanna, write an equation expressing U in terms of R.
Group Exercise
117. In Exercises 87–88, we used the data in a bar graph to develop linear functions that modeled trust in government and trust in media. For this group exercise, you might find it helpful to pattern your work after Exercises 87 and 88. Group members should begin by consulting an almanac, newspaper, magazine, or the Internet to find data that appear to lie approximately on or near a line. Working by hand or using a graphing utility, group members should construct scatter plots for the data that were assembled. If working by hand, draw a line that approximately fits the data in each scatter plot and then write its equation as a function in slope-intercept form. If using a graphing utility, obtain the equation of each regression line. Then use each linear function’s equation to make predictions about what might occur in the future. Are there circumstances that might affect the accuracy of the prediction? List some of these circumstances.
Retaining the Concepts
118. According to the U.S. Office of Management and Budget, the cost of maintaining existing public transportation infrastructure in 2020 was $103.4 billion and is projected to increase by $0.9 billion each year. By which year is the cost of maintaining existing public transportation infrastructure expected to reach $116 billion?
Exercises 121–123 will help you prepare for the material covered in the next section.
121. Write the slope-intercept form of the equation of the line passing through whose slope is the same as the line whose equation is .
122. Write an equation in general form of the line passing through whose slope is the negative reciprocal (the reciprocal with the opposite sign) of.
123. If , find
where and .
1.4: Exercise Set
1.4 Exercise Set
Practice Exercises
In Exercises 1–10, find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
1. (4, 7) and (8, 10)
2. (2, 1) and (3, 4)
3. and (2, 2)
4. and (2, 4)
5. and
6. and
7. and
8. and
9. (5, 3) and
10. and (3, 5)
In Exercises 11–38, use the given conditions to write an equation for each line in point-slope form and slope-intercept form.
11. , passing through (3, 5)
12. , passing through (1, 3)
13. , passing through
14. , passing through
15. , passing through
16. , passing through
17. , passing through
18. , passing through
19. , passing through
20. , passing through
21. , passing through the origin
22. , passing through the origin
23. , passing through
24. , passing through
25. Passing through (1, 2) and (5, 10)
26. Passing through (3, 5) and (8, 15)
27. Passing through and (0, 3)
28. Passing through and (0, 2)
29. Passing through and (2, 4)
30. Passing through and
31. Passing through and (3, 6)
32. Passing through and
33. Passing through and
34. Passing through and
35. Passing through (2, 4) with
36. Passing through with
37. and
38. and
In Exercises 39–48, give the slope and y-intercept of each line whose equation is given. Then graph the linear function.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
In Exercises 49–58, graph each equation in a rectangular coordinate system.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
In Exercises 59–66,
Rewrite the given equation in slope-intercept form.
Give the slope and y-intercept.
Use the slope and y-intercept to graph the linear function.
59.
60.
61.
62.
63.
64.
65.
66.
In Exercises 67–72, use intercepts to graph each equation.
67.
68.
69.
70.
71.
72.
Practice PLUS
In Exercises 73–76, find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical.
73. and
74. and
75. and
76. and
In Exercises 77–78, give the slope and y-intercept of each line whose equation is given. Assume that .
77.
78.
In Exercises 79–80, find the value of y if the line through the two given points is to have the indicated slope.
79. and (1, 4),
80. and ,
In Exercises 81–82, graph each linear function.
81.
82.
83. If one point on a line is and the line’s slope is , find the y-intercept.
84. If one point on a line is and the line’s slope is , find the y-intercept.
Use the figure to make the lists in Exercises 85–86.
85. List the slopes , and in order of decreasing size.
86. List the y-intercepts , and in order of decreasing size.
Application Exercises
Americans’ trust in government and the media has generally been on a downward trend since pollsters first asked questions on these topics in the second half of the twentieth century. Trust in government hit an all-time low of 14% in 2014, while trust in the media bottomed out at 32% in 2016. The bar graph shows the percentage of Americans trusting in the government and the media for five selected years. The data are displayed as two sets of five points each, one scatter plot for the percentage of Americans trusting in the government and one for the percentage of Americans trusting in the media. Also shown for each scatter plot is a line that passes through or near the five points. Use these lines to solve Exercises 87–88.
87. In this exercise, you will use the red line for trust in government shown on the scatter plot to develop a model for the percentage of Americans trusting in government.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in government, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in government in 2025. Round to the nearest percent.
88. In this exercise, you will use the blue line for trust in media shown on the scatter plot to develop a model for the percentage of Americans trusting in media.
Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of Americans trusting in media, y, x years after 2003. Round the slope to two decimal places.
Write the equation from part (a) in slope-intercept form. Use function notation.
Use the linear function to predict the percentage of Americans trusting in media in 2025. Round to the nearest percent.
The bar graph gives the life expectancy for American men and women born in six selected years. In Exercises 89–90, you will use the data to obtain models for life expectancy and make predictions about how long American men and women will live in the future.
89. Use the data for males shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent male life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show male life expectancies for 1980 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American men born x years after 1960.
Use the function from part (b) to project the life expectancy of American men born in 2020.
90. Use the data for females shown in the bar graph at the bottom of the previous column to solve this exercise.
Let x represent the number of birth years after 1960 and let y represent female life expectancy. Create a scatter plot that displays the data as a set of six points in a rectangular coordinate system.
Draw a line through the two points that show female life expectancies for 1970 and 2000. Use the coordinates of these points to write a linear function that models life expectancy, E(x), for American women born x years after 1960. Round the slope to two decimal places.
Use the function from part (b) to project the life expectancy of American women born in 2020.
91. Shown, again, is the scatter plot that indicates a relationship between the percentage of adult females in a country who are literate and the mortality of children under five. Also shown is a line that passes through or near the points. Find a linear function that models the data by finding the slope-intercept form of the line’s equation. Use the function to make a prediction about child mortality based on the percentage of adult females in a country who are literate.
92. Just as money doesn’t buy happiness for individuals, the two don’t necessarily go together for countries either. However, the scatter plot does show a relationship between a country’s annual per capita income and the percentage of people in that country who call themselves “happy.”
Source: Richard Layard, Happiness: Lessons from a New Science, Penguin, 2005
Draw a line that fits the data so that the spread of the data points around the line is as small as possible. Use the coordinates of two points along your line to write the slope-intercept form of its equation. Express the equation in function notation and use the linear function to make a prediction about national happiness based on per capita income.
Explaining the Concepts
93. What is the slope of a line and how is it found?
94. Describe how to write the equation of a line if the coordinates of two points along the line are known.
95. Explain how to derive the slope-intercept form of a line’s equation, , from the point-slope form
96. Explain how to graph the equation . Can this equation be expressed in slope-intercept form? Explain.
97. Explain how to use the general form of a line’s equation to find the line’s slope and y-intercept.
98. Explain how to use intercepts to graph the general form of a line’s equation.
99. Take another look at the scatter plot in Exercise 91. Although there is a relationship between literacy and child mortality, we cannot conclude that increased literacy causes child mortality to decrease. Offer two or more possible explanations for the data in the scatter plot.
Technology Exercises
Use a graphing utility to graph each equation in Exercises 100–103. Then use the feature to trace along the line and find the coordinates of two points. Use these points to compute the line’s slope. Check your result by using the coefficient of x in the line’s equation.
100.
101.
102.
103.
104. Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 developed countries.
Use the statistical menu of your graphing utility to enter the 19 ordered pairs of data items shown in the table.
Use the scatter plot capability to draw a scatter plot of the data.
Select the linear regression option. Use your utility to obtain values for a and b for the equation of the regression line, . You may also be given a correlation coefficient, r. Values of r close to 1 indicate that the points can be described by a linear relationship and the regression line has a positive slope. Values of r close to indicate that the points can be described by a linear relationship and the regression line has a negative slope. Values of r close to 0 indicate no linear relationship between the variables. In this case, a linear model does not accurately describe the data.
Use the appropriate sequence (consult your manual) to graph the regression equation on top of the points in the scatter plot.
Critical Thinking Exercises
Make Sense? In Exercises 105–108, determine whether each statement makes sense or does not make sense, and explain your reasoning.
105. The graph of my linear function at first increased, reached a maximum point, and then decreased.
106. A linear function that models tuition and fees at public four-year colleges from 2000 through 2020 has negative slope.
107. Because the variable m does not appear in , equations in this form make it impossible to determine the line’s slope.
108. The federal minimum wage was $7.25 per hour from 2009 through 2020, so models the minimum wage, , in dollars, for the domain .
In Exercises 109–112, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
109. The equation shows that no line can have a y-intercept that is numerically equal to its slope.
110. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.
111. The graph of the linear function is a line passing through the point (6, 0) with slope.
112. The graph of in the rectangular coordinate system is the single point (7, 0).
In Exercises 113–114, find the coefficients that must be placed in each shaded area so that the function’s graph will be a line satisfying the specified conditions.
113.
114.
115. Prove that the equation of a line passing through and can be written in the form . Why is this called the intercept form of a line?
116. Excited about the success of celebrity stamps, post office officials were rumored to have put forth a plan to institute two new types of thermometers. On these new scales, represents degrees Usher and represents degrees Rihanna. If it is known that , , and degrees Usher is linearly related to degrees Rihanna, write an equation expressing U in terms of R.
Group Exercise
117. In Exercises 87–88, we used the data in a bar graph to develop linear functions that modeled trust in government and trust in media. For this group exercise, you might find it helpful to pattern your work after Exercises 87 and 88. Group members should begin by consulting an almanac, newspaper, magazine, or the Internet to find data that appear to lie approximately on or near a line. Working by hand or using a graphing utility, group members should construct scatter plots for the data that were assembled. If working by hand, draw a line that approximately fits the data in each scatter plot and then write its equation as a function in slope-intercept form. If using a graphing utility, obtain the equation of each regression line. Then use each linear function’s equation to make predictions about what might occur in the future. Are there circumstances that might affect the accuracy of the prediction? List some of these circumstances.
Retaining the Concepts
118. According to the U.S. Office of Management and Budget, the cost of maintaining existing public transportation infrastructure in 2020 was $103.4 billion and is projected to increase by $0.9 billion each year. By which year is the cost of maintaining existing public transportation infrastructure expected to reach $116 billion?
As housing prices skyrocket, fewer U.S. young adults are buying homes and more are living with their parents. Figure 1.51 shows that in 2017, 38.4% of 25-to-34-year-old young adults owned homes, a decrease from the percentage displayed for 2000, and 22.0% lived with parents, nearly doubling the 2000 percentage.
Take a second look at Figure 1.51. The red graph is going down from left to right, indicating a negative rate of change in home ownership among young adults. The green graph is going up from left to right, indicating a positive rate of change in young adults living with parents. In this section, you will learn how to interpret slope as a rate of change. You will also explore the relationships between parallel and perpendicular lines.
Parallel and Perpendicular Lines
Two nonintersecting lines that lie in the same plane are parallel. If two lines do not intersect, the ratio of the vertical change to the horizontal change is the same for both lines. Because two parallel lines have the same “steepness,” they must have the same slope.
Slope and Parallel Lines
If two nonvertical lines are parallel, then they have the same slope.
If two distinct nonvertical lines have the same slope, then they are parallel.
Two distinct vertical lines, both with undefined slopes, are parallel.
Objective 1: Find slopes and equations of parallel and perpendicular lines
Objective 1 Find slopes and equations of parallel and perpendicular lines.
Watch Video
Example 1 Writing Equations of a Line Parallel to a Given Line
Write an equation of the line passing through and parallel to the line whose equation is . Express the equation in point-slope form and slope-intercept form.
Solution
The situation is illustrated in Figure 1.52. We are looking for the equation of the red line passing through and parallel to the blue line whose equation is . How do we obtain the equation of this red line? Notice that the line passes through the point . Using the point-slope form of the line’s equation, we have and .
With , the only thing missing from the equation of the red line is m, the slope. Do we know anything about the slope of either line in Figure 1.52? The answer is yes; we know the slope of the blue line on the right, whose equation is given.
Parallel lines have the same slope. Because the slope of the blue line is 2, the slope of the red line, the line whose equation we must write, is also We now have values for , and m for the red line.
The point-slope form of the red line’s equation is
Solving for y, we obtain the slope-intercept form of the equation.
Write an equation of the line passing through and parallel to the line whose equation is . Express the equation in point-slope form and slope-intercept form.
Two lines that intersect at a right angle are said to be perpendicular, shown in Figure 1.53. The relationship between the slopes of perpendicular lines is not as obvious as the relationship between parallel lines. Figure 1.53 shows line AB, with slope . Rotate line AB counterclockwise to the left to obtain line , perpendicular to line AB. The figure indicates that the rise and the run of the new line are reversed from the original line, but the former rise, the new run, is now negative. This means that the slope of the new line is. Notice that the product of the slopes of the two perpendicular lines is
This relationship holds for all perpendicular lines and is summarized in the box at the top of the next page.
Slope and Perpendicular Lines
If two nonvertical lines are perpendicular, then the product of their slopes is .
If the product of the slopes of two lines is , then the lines are perpendicular.
A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
An equivalent way of stating this relationship is to say that one line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other line. For example, if a line has slope 5, any line having slope is perpendicular to it. Similarly, if a line has slope, any line having slope is perpendicular to it.
Example 2 Writing Equations of a Line Perpendicular to a Given Line
Find the slope of any line that is perpendicular to the line whose equation is .
Write the equation of the line passing through and perpendicular to the line whose equation is . Express the equation in general form.
Solution
We begin by writing the equation of the given line, , in slope-intercept form. Solve for y.
The given line has slope. Any line perpendicular to this line has a slope that is the negative reciprocal of. Thus, the slope of any perpendicular line is 4.
Let’s begin by writing the point-slope form of the perpendicular line’s equation. Because the line passes through the point , we have and . In part (a), we determined that the slope of any line perpendicular to is 4, so the slope of this particular perpendicular line must also be 4: .
The point-slope form of the perpendicular line’s equation is
How can we express this equation, , in general form We need to obtain zero on one side of the equation. Let’s do this and keep A, the coefficient of x, positive.
In general form, the equation of the perpendicular line is .
Find the slope of any line that is perpendicular to the line whose equation is .
Write the equation of the line passing through and perpendicular to the line whose equation is . Express the equation in general form.
Objective 1: Find slopes and equations of parallel and perpendicular lines
Objective 1 Find slopes and equations of parallel and perpendicular lines.
Watch Video
Example 1 Writing Equations of a Line Parallel to a Given Line
Write an equation of the line passing through and parallel to the line whose equation is . Express the equation in point-slope form and slope-intercept form.
Solution
The situation is illustrated in Figure 1.52. We are looking for the equation of the red line passing through and parallel to the blue line whose equation is . How do we obtain the equation of this red line? Notice that the line passes through the point . Using the point-slope form of the line’s equation, we have and .
With , the only thing missing from the equation of the red line is m, the slope. Do we know anything about the slope of either line in Figure 1.52? The answer is yes; we know the slope of the blue line on the right, whose equation is given.
Parallel lines have the same slope. Because the slope of the blue line is 2, the slope of the red line, the line whose equation we must write, is also We now have values for , and m for the red line.
The point-slope form of the red line’s equation is
Solving for y, we obtain the slope-intercept form of the equation.
Write an equation of the line passing through and parallel to the line whose equation is . Express the equation in point-slope form and slope-intercept form.
Two lines that intersect at a right angle are said to be perpendicular, shown in Figure 1.53. The relationship between the slopes of perpendicular lines is not as obvious as the relationship between parallel lines. Figure 1.53 shows line AB, with slope . Rotate line AB counterclockwise to the left to obtain line , perpendicular to line AB. The figure indicates that the rise and the run of the new line are reversed from the original line, but the former rise, the new run, is now negative. This means that the slope of the new line is. Notice that the product of the slopes of the two perpendicular lines is
This relationship holds for all perpendicular lines and is summarized in the box at the top of the next page.
Slope and Perpendicular Lines
If two nonvertical lines are perpendicular, then the product of their slopes is .
If the product of the slopes of two lines is , then the lines are perpendicular.
A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
An equivalent way of stating this relationship is to say that one line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other line. For example, if a line has slope 5, any line having slope is perpendicular to it. Similarly, if a line has slope, any line having slope is perpendicular to it.
Example 2 Writing Equations of a Line Perpendicular to a Given Line
Find the slope of any line that is perpendicular to the line whose equation is .
Write the equation of the line passing through and perpendicular to the line whose equation is . Express the equation in general form.
Solution
We begin by writing the equation of the given line, , in slope-intercept form. Solve for y.
The given line has slope. Any line perpendicular to this line has a slope that is the negative reciprocal of. Thus, the slope of any perpendicular line is 4.
Let’s begin by writing the point-slope form of the perpendicular line’s equation. Because the line passes through the point , we have and . In part (a), we determined that the slope of any line perpendicular to is 4, so the slope of this particular perpendicular line must also be 4: .
The point-slope form of the perpendicular line’s equation is
How can we express this equation, , in general form We need to obtain zero on one side of the equation. Let’s do this and keep A, the coefficient of x, positive.
In general form, the equation of the perpendicular line is .
Find the slope of any line that is perpendicular to the line whose equation is .
Write the equation of the line passing through and perpendicular to the line whose equation is . Express the equation in general form.
Objective 1: Find slopes and equations of parallel and perpendicular lines
Objective 1 Find slopes and equations of parallel and perpendicular lines.
Watch Video
Example 1 Writing Equations of a Line Parallel to a Given Line
Write an equation of the line passing through and parallel to the line whose equation is . Express the equation in point-slope form and slope-intercept form.
Solution
The situation is illustrated in Figure 1.52. We are looking for the equation of the red line passing through and parallel to the blue line whose equation is . How do we obtain the equation of this red line? Notice that the line passes through the point . Using the point-slope form of the line’s equation, we have and .
With , the only thing missing from the equation of the red line is m, the slope. Do we know anything about the slope of either line in Figure 1.52? The answer is yes; we know the slope of the blue line on the right, whose equation is given.
Parallel lines have the same slope. Because the slope of the blue line is 2, the slope of the red line, the line whose equation we must write, is also We now have values for , and m for the red line.
The point-slope form of the red line’s equation is
Solving for y, we obtain the slope-intercept form of the equation.
Write an equation of the line passing through and parallel to the line whose equation is . Express the equation in point-slope form and slope-intercept form.
Two lines that intersect at a right angle are said to be perpendicular, shown in Figure 1.53. The relationship between the slopes of perpendicular lines is not as obvious as the relationship between parallel lines. Figure 1.53 shows line AB, with slope . Rotate line AB counterclockwise to the left to obtain line , perpendicular to line AB. The figure indicates that the rise and the run of the new line are reversed from the original line, but the former rise, the new run, is now negative. This means that the slope of the new line is. Notice that the product of the slopes of the two perpendicular lines is
This relationship holds for all perpendicular lines and is summarized in the box at the top of the next page.
Slope and Perpendicular Lines
If two nonvertical lines are perpendicular, then the product of their slopes is .
If the product of the slopes of two lines is , then the lines are perpendicular.
A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
An equivalent way of stating this relationship is to say that one line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other line. For example, if a line has slope 5, any line having slope is perpendicular to it. Similarly, if a line has slope, any line having slope is perpendicular to it.
Example 2 Writing Equations of a Line Perpendicular to a Given Line
Find the slope of any line that is perpendicular to the line whose equation is .
Write the equation of the line passing through and perpendicular to the line whose equation is . Express the equation in general form.
Solution
We begin by writing the equation of the given line, , in slope-intercept form. Solve for y.
The given line has slope. Any line perpendicular to this line has a slope that is the negative reciprocal of. Thus, the slope of any perpendicular line is 4.
Let’s begin by writing the point-slope form of the perpendicular line’s equation. Because the line passes through the point , we have and . In part (a), we determined that the slope of any line perpendicular to is 4, so the slope of this particular perpendicular line must also be 4: .
The point-slope form of the perpendicular line’s equation is
How can we express this equation, , in general form We need to obtain zero on one side of the equation. Let’s do this and keep A, the coefficient of x, positive.
In general form, the equation of the perpendicular line is .
Find the slope of any line that is perpendicular to the line whose equation is .
Write the equation of the line passing through and perpendicular to the line whose equation is . Express the equation in general form.
Objective 2: Interpret slope as rate of change
Slope as Rate of Change
Objective 2 Interpret slope as rate of change.
Watch Video
Slope is defined as the ratio of a change in y to a corresponding change in x. It describes how fast y is changing with respect to x. For a linear function, slope may be interpreted as the rate of change of the dependent variable per unit change in the independent variable.
Our next example shows how slope can be interpreted as a rate of change in an applied situation. When calculating slope in applied problems, keep track of the units in the numerator and the denominator.
Example 3 Slope as a Rate of Change
The line graphs for the living arrangements of young adults are shown again in Figure 1.54. Find the slope of the line segment for the percentage of young adults, age 25 to 34, owning a home. Describe what this slope represents.
We let x represent a year and y the percentage of young adults owning a home in that year. The two points shown on the line segment for home ownership have the following coordinates:
The slope indicates that the percentage of U.S. young adults, age 25 to 34, owning a home decreased at a rate of approximately 0.41 each year for the period from 2000 to 2017. The rate of change is per year.
Use the ordered pairs in Figure 1.54 on the previous page to find the slope of the green line segment for young adults, age 25 to 34, living with parents. Express the slope correct to two decimal places and describe what it represents.
Objective 3: Find a function’s average rate of change
The Average Rate of Change of a Function
Objective 3 Find a function’s average rate of change.
Watch Video
If the graph of a function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line. For example, Figure 1.55 shows the graph of a particular man’s height, in inches, as a function of his age, in years. Two points on the graph are labeled: (13, 57) and (18, 76). At age 13, this man was 57 inches tall and at age 18, he was 76 inches tall.
The man’s average growth rate between ages 13 and 18 is the slope of the secant line containing (13, 57) and (18, 76):
This man’s average rate of change, or average growth rate, from age 13 to age 18 was , or 3.8, inches per year.
The Average Rate of Change of a Function
Let and be distinct points on the graph of a function f. (See Figure 1.56.) The average rate of change of f from to denoted by (read “delta y divided by delta x” or “change in y divided by change in x”), is
Figure 1.57(a) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (0, 1).
Figure 1.57(a) The secant line of from to
The average rate of change of from to is
Figure 1.57(b) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (1, 2). Can you see that the graph rises more steeply on the interval (1, 2) than on (0, 1)? This is because the average rate of change from to is greater than the average rate of change from to .
Figure 1.57(b) The secant line of from to
The average rate of change of from to is
Figure 1.57(c) shows the secant line of from to . The average rate of change is negative and the function is decreasing on the interval .
Suppose we are interested in the average rate of change of f from to . In this case, the average rate of change is
Do you recognize the last expression? It is the difference quotient that you used in Section 1.3. Thus, the difference quotient gives the average rate of change of a function from x to In the difference quotient, h is thought of as a number very close to 0. In this way, the average rate of change can be found for a very short interval.
Example 5 Finding the Average Rate of Change
When a person receives a drug injected into a muscle, the concentration of the drug in the body, measured in milligrams per 100 milliliters, is a function of the time elapsed after the injection, measured in hours. Figure 1.58 shows the graph of such a function, where x represents hours after the injection and is the drug’s concentration at time x. Find the average rate of change in the drug’s concentration between 3 and 7 hours.
Figure 1.58 Concentration of a drug as a function of time
At 3 hours, the drug’s concentration is 0.05 and at 7 hours, the concentration is 0.02. The average rate of change in its concentration between 3 and 7 hours is
The average rate of change is . This means that the drug’s concentration is decreasing at an average rate of 0.0075 milligram per 100 milliliters per hour.
Use Figure 1.58 to find the average rate of change in the drug’s concentration between 1 hour and 3 hours.
The average velocity of an object is its change in position divided by the change in time between the starting and ending positions. If a function expresses an object’s position in terms of time, the function’s average rate of change describes the object’s average velocity.
Average Velocity of an Object
Suppose that a function expresses an object’s position, , in terms of time, . The average velocity of the object from to is
Example 6 Finding Average Velocity
The distance, , in feet, traveled by a ball rolling down a ramp is given by the function
where is the time, in seconds, after the ball is released. Find the ball’s average velocity from
to .
to .
to .
Solution
The ball’s average velocity between 2 and 3 seconds is
The ball’s average velocity between 2 and 2.5 seconds is
The ball’s average velocity between 2 and 2.01 seconds is
In Example 6, observe that each calculation begins at 2 seconds and involves shorter and shorter time intervals. In calculus, this procedure leads to the concept of instantaneous, as opposed to average, velocity. Instantaneous velocity is discussed in the introduction to calculus in Chapter 11.
The distance, , in feet, traveled by a ball rolling down a ramp is given by the function
where is the time, in seconds, after the ball is released. Find the ball’s average velocity from
to .
to .
to .
Objective 3: Find a function’s average rate of change
The Average Rate of Change of a Function
Objective 3 Find a function’s average rate of change.
Watch Video
If the graph of a function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line. For example, Figure 1.55 shows the graph of a particular man’s height, in inches, as a function of his age, in years. Two points on the graph are labeled: (13, 57) and (18, 76). At age 13, this man was 57 inches tall and at age 18, he was 76 inches tall.
The man’s average growth rate between ages 13 and 18 is the slope of the secant line containing (13, 57) and (18, 76):
This man’s average rate of change, or average growth rate, from age 13 to age 18 was , or 3.8, inches per year.
The Average Rate of Change of a Function
Let and be distinct points on the graph of a function f. (See Figure 1.56.) The average rate of change of f from to denoted by (read “delta y divided by delta x” or “change in y divided by change in x”), is
Figure 1.57(a) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (0, 1).
Figure 1.57(a) The secant line of from to
The average rate of change of from to is
Figure 1.57(b) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (1, 2). Can you see that the graph rises more steeply on the interval (1, 2) than on (0, 1)? This is because the average rate of change from to is greater than the average rate of change from to .
Figure 1.57(b) The secant line of from to
The average rate of change of from to is
Figure 1.57(c) shows the secant line of from to . The average rate of change is negative and the function is decreasing on the interval .
Suppose we are interested in the average rate of change of f from to . In this case, the average rate of change is
Do you recognize the last expression? It is the difference quotient that you used in Section 1.3. Thus, the difference quotient gives the average rate of change of a function from x to In the difference quotient, h is thought of as a number very close to 0. In this way, the average rate of change can be found for a very short interval.
Example 5 Finding the Average Rate of Change
When a person receives a drug injected into a muscle, the concentration of the drug in the body, measured in milligrams per 100 milliliters, is a function of the time elapsed after the injection, measured in hours. Figure 1.58 shows the graph of such a function, where x represents hours after the injection and is the drug’s concentration at time x. Find the average rate of change in the drug’s concentration between 3 and 7 hours.
Figure 1.58 Concentration of a drug as a function of time
At 3 hours, the drug’s concentration is 0.05 and at 7 hours, the concentration is 0.02. The average rate of change in its concentration between 3 and 7 hours is
The average rate of change is . This means that the drug’s concentration is decreasing at an average rate of 0.0075 milligram per 100 milliliters per hour.
Use Figure 1.58 to find the average rate of change in the drug’s concentration between 1 hour and 3 hours.
The average velocity of an object is its change in position divided by the change in time between the starting and ending positions. If a function expresses an object’s position in terms of time, the function’s average rate of change describes the object’s average velocity.
Average Velocity of an Object
Suppose that a function expresses an object’s position, , in terms of time, . The average velocity of the object from to is
Example 6 Finding Average Velocity
The distance, , in feet, traveled by a ball rolling down a ramp is given by the function
where is the time, in seconds, after the ball is released. Find the ball’s average velocity from
to .
to .
to .
Solution
The ball’s average velocity between 2 and 3 seconds is
The ball’s average velocity between 2 and 2.5 seconds is
The ball’s average velocity between 2 and 2.01 seconds is
In Example 6, observe that each calculation begins at 2 seconds and involves shorter and shorter time intervals. In calculus, this procedure leads to the concept of instantaneous, as opposed to average, velocity. Instantaneous velocity is discussed in the introduction to calculus in Chapter 11.
The distance, , in feet, traveled by a ball rolling down a ramp is given by the function
where is the time, in seconds, after the ball is released. Find the ball’s average velocity from
to .
to .
to .
Objective 3: Find a function’s average rate of change
The Average Rate of Change of a Function
Objective 3 Find a function’s average rate of change.
Watch Video
If the graph of a function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line. For example, Figure 1.55 shows the graph of a particular man’s height, in inches, as a function of his age, in years. Two points on the graph are labeled: (13, 57) and (18, 76). At age 13, this man was 57 inches tall and at age 18, he was 76 inches tall.
The man’s average growth rate between ages 13 and 18 is the slope of the secant line containing (13, 57) and (18, 76):
This man’s average rate of change, or average growth rate, from age 13 to age 18 was , or 3.8, inches per year.
The Average Rate of Change of a Function
Let and be distinct points on the graph of a function f. (See Figure 1.56.) The average rate of change of f from to denoted by (read “delta y divided by delta x” or “change in y divided by change in x”), is
Figure 1.57(a) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (0, 1).
Figure 1.57(a) The secant line of from to
The average rate of change of from to is
Figure 1.57(b) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (1, 2). Can you see that the graph rises more steeply on the interval (1, 2) than on (0, 1)? This is because the average rate of change from to is greater than the average rate of change from to .
Figure 1.57(b) The secant line of from to
The average rate of change of from to is
Figure 1.57(c) shows the secant line of from to . The average rate of change is negative and the function is decreasing on the interval .
Suppose we are interested in the average rate of change of f from to . In this case, the average rate of change is
Do you recognize the last expression? It is the difference quotient that you used in Section 1.3. Thus, the difference quotient gives the average rate of change of a function from x to In the difference quotient, h is thought of as a number very close to 0. In this way, the average rate of change can be found for a very short interval.
Example 5 Finding the Average Rate of Change
When a person receives a drug injected into a muscle, the concentration of the drug in the body, measured in milligrams per 100 milliliters, is a function of the time elapsed after the injection, measured in hours. Figure 1.58 shows the graph of such a function, where x represents hours after the injection and is the drug’s concentration at time x. Find the average rate of change in the drug’s concentration between 3 and 7 hours.
Figure 1.58 Concentration of a drug as a function of time
At 3 hours, the drug’s concentration is 0.05 and at 7 hours, the concentration is 0.02. The average rate of change in its concentration between 3 and 7 hours is
The average rate of change is . This means that the drug’s concentration is decreasing at an average rate of 0.0075 milligram per 100 milliliters per hour.
Use Figure 1.58 to find the average rate of change in the drug’s concentration between 1 hour and 3 hours.
The average velocity of an object is its change in position divided by the change in time between the starting and ending positions. If a function expresses an object’s position in terms of time, the function’s average rate of change describes the object’s average velocity.
Average Velocity of an Object
Suppose that a function expresses an object’s position, , in terms of time, . The average velocity of the object from to is
Example 6 Finding Average Velocity
The distance, , in feet, traveled by a ball rolling down a ramp is given by the function
where is the time, in seconds, after the ball is released. Find the ball’s average velocity from
to .
to .
to .
Solution
The ball’s average velocity between 2 and 3 seconds is
The ball’s average velocity between 2 and 2.5 seconds is
The ball’s average velocity between 2 and 2.01 seconds is
In Example 6, observe that each calculation begins at 2 seconds and involves shorter and shorter time intervals. In calculus, this procedure leads to the concept of instantaneous, as opposed to average, velocity. Instantaneous velocity is discussed in the introduction to calculus in Chapter 11.
The distance, , in feet, traveled by a ball rolling down a ramp is given by the function
where is the time, in seconds, after the ball is released. Find the ball’s average velocity from
to .
to .
to .
Objective 3: Find a function’s average rate of change
The Average Rate of Change of a Function
Objective 3 Find a function’s average rate of change.
Watch Video
If the graph of a function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line. For example, Figure 1.55 shows the graph of a particular man’s height, in inches, as a function of his age, in years. Two points on the graph are labeled: (13, 57) and (18, 76). At age 13, this man was 57 inches tall and at age 18, he was 76 inches tall.
The man’s average growth rate between ages 13 and 18 is the slope of the secant line containing (13, 57) and (18, 76):
This man’s average rate of change, or average growth rate, from age 13 to age 18 was , or 3.8, inches per year.
The Average Rate of Change of a Function
Let and be distinct points on the graph of a function f. (See Figure 1.56.) The average rate of change of f from to denoted by (read “delta y divided by delta x” or “change in y divided by change in x”), is
Figure 1.57(a) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (0, 1).
Figure 1.57(a) The secant line of from to
The average rate of change of from to is
Figure 1.57(b) shows the secant line of from to . The average rate of change is positive and the function is increasing on the interval (1, 2). Can you see that the graph rises more steeply on the interval (1, 2) than on (0, 1)? This is because the average rate of change from to is greater than the average rate of change from to .
Figure 1.57(b) The secant line of from to
The average rate of change of from to is
Figure 1.57(c) shows the secant line of from to . The average rate of change is negative and the function is decreasing on the interval .
Suppose we are interested in the average rate of change of f from to . In this case, the average rate of change is
Do you recognize the last expression? It is the difference quotient that you used in Section 1.3. Thus, the difference quotient gives the average rate of change of a function from x to In the difference quotient, h is thought of as a number very close to 0. In this way, the average rate of change can be found for a very short interval.
Example 5 Finding the Average Rate of Change
When a person receives a drug injected into a muscle, the concentration of the drug in the body, measured in milligrams per 100 milliliters, is a function of the time elapsed after the injection, measured in hours. Figure 1.58 shows the graph of such a function, where x represents hours after the injection and is the drug’s concentration at time x. Find the average rate of change in the drug’s concentration between 3 and 7 hours.
Figure 1.58 Concentration of a drug as a function of time
At 3 hours, the drug’s concentration is 0.05 and at 7 hours, the concentration is 0.02. The average rate of change in its concentration between 3 and 7 hours is
The average rate of change is . This means that the drug’s concentration is decreasing at an average rate of 0.0075 milligram per 100 milliliters per hour.
Use Figure 1.58 to find the average rate of change in the drug’s concentration between 1 hour and 3 hours.
The average velocity of an object is its change in position divided by the change in time between the starting and ending positions. If a function expresses an object’s position in terms of time, the function’s average rate of change describes the object’s average velocity.
Average Velocity of an Object
Suppose that a function expresses an object’s position, , in terms of time, . The average velocity of the object from to is
Example 6 Finding Average Velocity
The distance, , in feet, traveled by a ball rolling down a ramp is given by the function
where is the time, in seconds, after the ball is released. Find the ball’s average velocity from
to .
to .
to .
Solution
The ball’s average velocity between 2 and 3 seconds is
The ball’s average velocity between 2 and 2.5 seconds is
The ball’s average velocity between 2 and 2.01 seconds is
In Example 6, observe that each calculation begins at 2 seconds and involves shorter and shorter time intervals. In calculus, this procedure leads to the concept of instantaneous, as opposed to average, velocity. Instantaneous velocity is discussed in the introduction to calculus in Chapter 11.
In Exercises 5–8, use the given conditions to write an equation for each line in point-slope form and slope-intercept form.
5. Passing through and parallel to the line whose equation is
6. Passing through and parallel to the line whose equation is
7. Passing through and perpendicular to the line whose equation is
8. Passing through and perpendicular to the line whose equation is
In Exercises 9–12, use the given conditions to write an equation for each line in point-slope form and general form.
9. Passing through and parallel to the line whose equation is
10. Passing through and parallel to the line whose equation is
11. Passing through and perpendicular to the line whose equation is
12. Passing through and perpendicular to the line whose equation is
In Exercises 13–18, find the average rate of change of the function from to .
13. from to
14. from to
15. from to
16. from to
17. from to
18. from to
In Exercises 19–20, suppose that a ball is rolling down a ramp. The distance traveled by the ball is given by the function in each exercise, where t is the time, in seconds, after the ball is released, and is measured in feet. For each given function, find the ball’s average velocity from
.
.
.
.
19.
20.
Practice PLUS
In Exercises 21–26, write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions.
21. The graph of f passes through and is perpendicular to the line whose equation is .
22. The graph of f passes through and is perpendicular to the line whose equation is .
23. The graph of f passes through and is perpendicular to the line that has an x-intercept of 2 and a y-intercept of .
24. The graph of f passes through and is perpendicular to the line that has an x-intercept of 3 and a y-intercept of .
25. The graph of f is perpendicular to the line whose equation is and has the same y-intercept as this line.
26. The graph of f is perpendicular to the line whose equation is and has the same y-intercept as this line.
Application Exercises
The bar graph shows that as costs changed over the decades, Americans devoted less of their budget to groceries and more to health care.
In Exercises 27–28, find a linear function in slope-intercept form that models the given description. Each function should model the percentage of total spending, p(x), by Americans x years after 1950.
27. In 1950, Americans spent 22% of their budget on food. This has decreased at an average rate of approximately 0.25% per year since then.
28. In 1950, Americans spent 3% of their budget on health care. This has increased at an average rate of approximately 0.22% per year since then.
The stated intent of the 1994 “don’t ask, don’t tell” policy was to reduce the number of LGBTQ discharges from the military. Nearly 14,000 active-duty LGBTQ servicemembers were dismissed under the policy, which officially ended in 2011, after 18 years. The line graph shows the number of discharges under “don’t ask, don’t tell” from 1994 through 2010. Use the data displayed by the graph to solve Exercises 29–30.
(In Exercises 29–30, be sure to refer to the graph at the bottom of the previous page.)
29. Find the average rate of change, rounded to the nearest whole number, from 1994 through 1998. Describe what this means.
30. Find the average rate of change, rounded to the nearest whole number, from 2001 through 2006. Describe what this means.
The function models the number of discharges, , under “don’t ask, don’t tell” x years after 1994. Use this model and its graph, shown below, to solve Exercises 31–32.
Find the slope of the secant line, rounded to the nearest whole number, from to .
Does the slope from part (a) underestimate or overestimate the average yearly increase that you determined in Exercise 29? By how much?
32.
Find the slope of the secant line, rounded to the nearest whole number, from to .
Does the slope from part (b) underestimate or overestimate the average yearly decrease that you determined in Exercise 30? By how much?
Explaining the Concepts
33. If two lines are parallel, describe the relationship between their slopes.
34. If two lines are perpendicular, describe the relationship between their slopes.
35. If you know a point on a line and you know the equation of a line perpendicular to this line, explain how to write the line’s equation.
36. A formula in the form models the average retail price, y, of a new car x years after 2000. Would you expect m to be positive, negative, or zero? Explain your answer.
37. What is a secant line?
38. What is the average rate of change of a function?
Technology Exercises
39.
Why are the lines whose equations are and perpendicular?
Use a graphing utility to graph the equations in a by viewing rectangle. Do the lines appear to be perpendicular?
Now use the zoom square feature of your utility. Describe what happens to the graphs. Explain why this is so.
Critical Thinking Exercises
Make Sense? In Exercises 40–43, determine whether each statement makes sense or does not make sense, and explain your reasoning.
40. I computed the slope of one line to be and the slope of a second line to be , so the lines must be perpendicular.
41. I have linear functions that model changes for men and women over the same time period. The functions have the same slope, so their graphs are parallel lines, indicating that the rate of change for men is the same as the rate of change for women.
42. The graph of my function is not a straight line, so I cannot use slope to analyze its rates of change.
43. According to the Blitzer Bonus here, calculus studies change by analyzing slopes of secant lines over successively shorter intervals.
44. What is the slope of a line that is perpendicular to the line whose equation is and
45. Determine the value of A so that the line whose equation is is perpendicular to the line containing the points and .
47. After a 30% price reduction, you purchase a 50 ˝ 4K UHD TV for $245. What was the television’s price before the reduction? (Section P.8, Example 4)
Exercises 49–51 will help you prepare for the material covered in the next section. In each exercise, graph the functions in parts (a) and (b) in the same rectangular coordinate system.
49.
Graph using the ordered pairs , , and .
Subtract 4 from each y-coordinate of the ordered pairs in part (a). Then graph the ordered pairs and connect them with two linear pieces.
Describe the relationship between the graph in part (b) and the graph in part (a).
50.
Graph using the ordered pairs , , and .
Add 2 to each x-coordinate of the ordered pairs in part (a). Then graph the ordered pairs and connect them with a smooth curve.
Describe the relationship between the graph in part (b) and the graph in part (a).
51.
Graph using the ordered pairs , , and .
Replace each x-coordinate of the ordered pairs in part (a) with its opposite, or additive inverse. Then graph the ordered pairs and connect them with a smooth curve.
Describe the relationship between the graph in part (b) and the graph in part (a).
1.5: Exercise Set
1.5 Exercise Set
Practice Exercises
In Exercises 1–4, write an equation for line L in point-slope form and slope-intercept form.
In Exercises 5–8, use the given conditions to write an equation for each line in point-slope form and slope-intercept form.
5. Passing through and parallel to the line whose equation is
6. Passing through and parallel to the line whose equation is
7. Passing through and perpendicular to the line whose equation is
8. Passing through and perpendicular to the line whose equation is
In Exercises 9–12, use the given conditions to write an equation for each line in point-slope form and general form.
9. Passing through and parallel to the line whose equation is
10. Passing through and parallel to the line whose equation is
11. Passing through and perpendicular to the line whose equation is
12. Passing through and perpendicular to the line whose equation is
In Exercises 13–18, find the average rate of change of the function from to .
13. from to
14. from to
15. from to
16. from to
17. from to
18. from to
In Exercises 19–20, suppose that a ball is rolling down a ramp. The distance traveled by the ball is given by the function in each exercise, where t is the time, in seconds, after the ball is released, and is measured in feet. For each given function, find the ball’s average velocity from
.
.
.
.
19.
20.
Practice PLUS
In Exercises 21–26, write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions.
21. The graph of f passes through and is perpendicular to the line whose equation is .
22. The graph of f passes through and is perpendicular to the line whose equation is .
23. The graph of f passes through and is perpendicular to the line that has an x-intercept of 2 and a y-intercept of .
24. The graph of f passes through and is perpendicular to the line that has an x-intercept of 3 and a y-intercept of .
25. The graph of f is perpendicular to the line whose equation is and has the same y-intercept as this line.
26. The graph of f is perpendicular to the line whose equation is and has the same y-intercept as this line.
Application Exercises
The bar graph shows that as costs changed over the decades, Americans devoted less of their budget to groceries and more to health care.
In Exercises 27–28, find a linear function in slope-intercept form that models the given description. Each function should model the percentage of total spending, p(x), by Americans x years after 1950.
27. In 1950, Americans spent 22% of their budget on food. This has decreased at an average rate of approximately 0.25% per year since then.
28. In 1950, Americans spent 3% of their budget on health care. This has increased at an average rate of approximately 0.22% per year since then.
The stated intent of the 1994 “don’t ask, don’t tell” policy was to reduce the number of LGBTQ discharges from the military. Nearly 14,000 active-duty LGBTQ servicemembers were dismissed under the policy, which officially ended in 2011, after 18 years. The line graph shows the number of discharges under “don’t ask, don’t tell” from 1994 through 2010. Use the data displayed by the graph to solve Exercises 29–30.
(In Exercises 29–30, be sure to refer to the graph at the bottom of the previous page.)
29. Find the average rate of change, rounded to the nearest whole number, from 1994 through 1998. Describe what this means.
30. Find the average rate of change, rounded to the nearest whole number, from 2001 through 2006. Describe what this means.
The function models the number of discharges, , under “don’t ask, don’t tell” x years after 1994. Use this model and its graph, shown below, to solve Exercises 31–32.
Find the slope of the secant line, rounded to the nearest whole number, from to .
Does the slope from part (a) underestimate or overestimate the average yearly increase that you determined in Exercise 29? By how much?
32.
Find the slope of the secant line, rounded to the nearest whole number, from to .
Does the slope from part (b) underestimate or overestimate the average yearly decrease that you determined in Exercise 30? By how much?
Explaining the Concepts
33. If two lines are parallel, describe the relationship between their slopes.
34. If two lines are perpendicular, describe the relationship between their slopes.
35. If you know a point on a line and you know the equation of a line perpendicular to this line, explain how to write the line’s equation.
36. A formula in the form models the average retail price, y, of a new car x years after 2000. Would you expect m to be positive, negative, or zero? Explain your answer.
37. What is a secant line?
38. What is the average rate of change of a function?
Technology Exercises
39.
Why are the lines whose equations are and perpendicular?
Use a graphing utility to graph the equations in a by viewing rectangle. Do the lines appear to be perpendicular?
Now use the zoom square feature of your utility. Describe what happens to the graphs. Explain why this is so.
Critical Thinking Exercises
Make Sense? In Exercises 40–43, determine whether each statement makes sense or does not make sense, and explain your reasoning.
40. I computed the slope of one line to be and the slope of a second line to be , so the lines must be perpendicular.
41. I have linear functions that model changes for men and women over the same time period. The functions have the same slope, so their graphs are parallel lines, indicating that the rate of change for men is the same as the rate of change for women.
42. The graph of my function is not a straight line, so I cannot use slope to analyze its rates of change.
43. According to the Blitzer Bonus here, calculus studies change by analyzing slopes of secant lines over successively shorter intervals.
44. What is the slope of a line that is perpendicular to the line whose equation is and
45. Determine the value of A so that the line whose equation is is perpendicular to the line containing the points and .
47. After a 30% price reduction, you purchase a 50 ˝ 4K UHD TV for $245. What was the television’s price before the reduction? (Section P.8, Example 4)
Exercises 49–51 will help you prepare for the material covered in the next section. In each exercise, graph the functions in parts (a) and (b) in the same rectangular coordinate system.
49.
Graph using the ordered pairs , , and .
Subtract 4 from each y-coordinate of the ordered pairs in part (a). Then graph the ordered pairs and connect them with two linear pieces.
Describe the relationship between the graph in part (b) and the graph in part (a).
50.
Graph using the ordered pairs , , and .
Add 2 to each x-coordinate of the ordered pairs in part (a). Then graph the ordered pairs and connect them with a smooth curve.
Describe the relationship between the graph in part (b) and the graph in part (a).
51.
Graph using the ordered pairs , , and .
Replace each x-coordinate of the ordered pairs in part (a) with its opposite, or additive inverse. Then graph the ordered pairs and connect them with a smooth curve.
Describe the relationship between the graph in part (b) and the graph in part (a).
Chapter 1: Mid-Chapter Check Point
Chapter 1 Mid-Chapter Check Point
What You Know: We learned that a function is a relation in which no two ordered pairs have the same first component and different second components. We represented functions as equations and used function notation. We graphed functions and applied the vertical line test to identify graphs of functions. We determined the domain and range of a function from its graph, using inputs on the x-axis for the domain and outputs on the y-axis for the range. We used graphs to identify intervals on which functions increase, decrease, or are constant, as well as to locate relative maxima or minima. We determined when graphs of equations are symmetric with respect to the y-axis (no change when is substituted for x), the x-axis (no change when is substituted for y), and the origin (no change when is substituted for x and is substituted for y). We identified even functions [ y-axis symmetry] and odd functions [ origin symmetry]. Finally, we studied linear functions and slope, using slope (change in y divided by change in x) to develop various forms for equations of lines:
We saw that parallel lines have the same slope and that perpendicular lines have slopes that are negative reciprocals. For linear functions, slope was interpreted as the rate of change of the dependent variable per unit change in the independent variable. For nonlinear functions, the slope of the secant line between and described the average rate of change of f from to
In Exercises 1–6, determine whether each relation is a function. Give the domain and range for each relation.
1.
2.
3.
4.
5.
6.
In Exercises 7–8, determine whether each equation defines y as a function of x.
7.
8.
Use the graph of f to solve Exercises 9–24. Where applicable, use interval notation.
9. Explain why f represents the graph of a function.
10. Find the domain of f.
11. Find the range of f.
12. Find the x-intercept(s).
13. Find the y-intercept.
14. Find the interval(s) on which f is increasing.
15. Find the interval(s) on which f is decreasing.
16. At what number does f have a relative maximum?
17. What is the relative maximum of f?
18. Find .
19. For what value or values of x is
20. For what value or values of x is
21. For what values of x is
22. Is positive or negative?
23. Is f even, odd, or neither?
24. Find the average rate of change of f from to .
In Exercises 25–26, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
25.
26.
In Exercises 27–38, graph each equation in a rectangular coordinate system.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39. Let .
Find . Is f even, odd, or neither?
Find .
40. Let
Find .
Find .
In Exercises 41–44, write a function in slope-intercept form whose graph satisfies the given conditions.
41. , passing through
42. Passing through and (2, 1)
43. Passing through and parallel to the line whose equation is
44. Passing through and perpendicular to the line whose equation is
45. Determine whether the line through and (7, 0) is parallel to a second line through and (1, 6).
46. Exercise is useful not only in preventing depression, but also as a treatment. The following graphs show the percentage of patients with depression in remission when exercise (brisk walking) was used as a treatment. (The control group that engaged in no exercise had 11% of the patients in remission.)
Find the slope of the line passing through the two points shown by the voice balloons. Express the slope as a decimal.
Use your answer from part (a) to complete this statement:
For each minute of brisk walking, the percentage of patients with depression in remission increased by _____%. The rate of change is _____% per _____________.
47. Find the average rate of change of from to .
Chapter 1: Mid-Chapter Check Point
Chapter 1 Mid-Chapter Check Point
What You Know: We learned that a function is a relation in which no two ordered pairs have the same first component and different second components. We represented functions as equations and used function notation. We graphed functions and applied the vertical line test to identify graphs of functions. We determined the domain and range of a function from its graph, using inputs on the x-axis for the domain and outputs on the y-axis for the range. We used graphs to identify intervals on which functions increase, decrease, or are constant, as well as to locate relative maxima or minima. We determined when graphs of equations are symmetric with respect to the y-axis (no change when is substituted for x), the x-axis (no change when is substituted for y), and the origin (no change when is substituted for x and is substituted for y). We identified even functions [ y-axis symmetry] and odd functions [ origin symmetry]. Finally, we studied linear functions and slope, using slope (change in y divided by change in x) to develop various forms for equations of lines:
We saw that parallel lines have the same slope and that perpendicular lines have slopes that are negative reciprocals. For linear functions, slope was interpreted as the rate of change of the dependent variable per unit change in the independent variable. For nonlinear functions, the slope of the secant line between and described the average rate of change of f from to
In Exercises 1–6, determine whether each relation is a function. Give the domain and range for each relation.
1.
2.
3.
4.
5.
6.
In Exercises 7–8, determine whether each equation defines y as a function of x.
7.
8.
Use the graph of f to solve Exercises 9–24. Where applicable, use interval notation.
9. Explain why f represents the graph of a function.
10. Find the domain of f.
11. Find the range of f.
12. Find the x-intercept(s).
13. Find the y-intercept.
14. Find the interval(s) on which f is increasing.
15. Find the interval(s) on which f is decreasing.
16. At what number does f have a relative maximum?
17. What is the relative maximum of f?
18. Find .
19. For what value or values of x is
20. For what value or values of x is
21. For what values of x is
22. Is positive or negative?
23. Is f even, odd, or neither?
24. Find the average rate of change of f from to .
In Exercises 25–26, determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
25.
26.
In Exercises 27–38, graph each equation in a rectangular coordinate system.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39. Let .
Find . Is f even, odd, or neither?
Find .
40. Let
Find .
Find .
In Exercises 41–44, write a function in slope-intercept form whose graph satisfies the given conditions.
41. , passing through
42. Passing through and (2, 1)
43. Passing through and parallel to the line whose equation is
44. Passing through and perpendicular to the line whose equation is
45. Determine whether the line through and (7, 0) is parallel to a second line through and (1, 6).
46. Exercise is useful not only in preventing depression, but also as a treatment. The following graphs show the percentage of patients with depression in remission when exercise (brisk walking) was used as a treatment. (The control group that engaged in no exercise had 11% of the patients in remission.)
Find the slope of the line passing through the two points shown by the voice balloons. Express the slope as a decimal.
Use your answer from part (a) to complete this statement:
For each minute of brisk walking, the percentage of patients with depression in remission increased by _____%. The rate of change is _____% per _____________.
47. Find the average rate of change of from to .
Objective 1: Recognize graphs of common functions
Graphs of Common Functions
Objective 1 Recognize graphs of common functions.
Watch Video
Table 1.4 gives names to seven frequently encountered functions in algebra. The table shows each function’s graph and lists characteristics of the function. Study the shape of each graph and take a few minutes to verify the function’s characteristics from its graph. Knowing these graphs is essential for analyzing their transformations into more complicated graphs.
Objective 2: Use vertical shifts to graph functions
Vertical Shifts
Objective 2 Use vertical shifts to graph functions.
Watch Video
Let’s begin by looking at three graphs whose shapes are the same. Figure 1.60 shows the graphs. The black graph in the middle is the standard quadratic function, . Now, look at the blue graph on the top. The equation of this graph, , adds 2 to the right side of . The y-coordinate of each point of g is 2 more than the corresponding y-coordinate of each point of f. What effect does this have on the graph of f? It shifts the graph vertically up by 2 units.
Finally, look at the red graph on the bottom in Figure 1.60. The equation of this graph, , subtracts 3 from the right side of . The y-coordinate of each point of h is 3 less than the corresponding y-coordinate of each point of f. What effect does this have on the graph of f? It shifts the graph vertically down by 3 units.
In general, if c is positive, shifts the graph of f upward c units and shifts the graph of f downward c units. These are called vertical shifts of the graph of f.
Vertical Shifts
Let f be a function and c a positive real number.
The graph of is the graph of shifted c units vertically upward.
The graph of is the graph of shifted c units vertically downward.
Objective 2: Use vertical shifts to graph functions
Vertical Shifts
Objective 2 Use vertical shifts to graph functions.
Watch Video
Let’s begin by looking at three graphs whose shapes are the same. Figure 1.60 shows the graphs. The black graph in the middle is the standard quadratic function, . Now, look at the blue graph on the top. The equation of this graph, , adds 2 to the right side of . The y-coordinate of each point of g is 2 more than the corresponding y-coordinate of each point of f. What effect does this have on the graph of f? It shifts the graph vertically up by 2 units.
Finally, look at the red graph on the bottom in Figure 1.60. The equation of this graph, , subtracts 3 from the right side of . The y-coordinate of each point of h is 3 less than the corresponding y-coordinate of each point of f. What effect does this have on the graph of f? It shifts the graph vertically down by 3 units.
In general, if c is positive, shifts the graph of f upward c units and shifts the graph of f downward c units. These are called vertical shifts of the graph of f.
Vertical Shifts
Let f be a function and c a positive real number.
The graph of is the graph of shifted c units vertically upward.
The graph of is the graph of shifted c units vertically downward.
Objective 3: Use horizontal shifts to graph functions
Horizontal Shifts
Objective 3 Use horizontal shifts to graph functions.
Watch Video
We return to the graph of , the standard quadratic function. In Figure 1.61, the graph of function f is in the middle of the three graphs. By contrast to the vertical shift situation, this time there are graphs to the left and to the right of the graph of f. Look at the blue graph on the right. The equation of this graph, , subtracts 3 from each value of x before squaring it. What effect does this have on the graph of It shifts the graph horizontally to the right by 3 units.
Does it seem strange that subtracting 3 in the domain causes a shift of 3 units to the right? Perhaps a partial table of coordinates for each function will numerically convince you of this shift.
x
0
1
2
x
1
2
3
4
5
Notice that for the values of and g(x) to be the same, the values of x used in graphing g must each be 3 units greater than those used to graph f. For this reason, the graph of g is the graph of f shifted 3 units to the right.
Now, look at the red graph on the left in Figure 1.61. The equation of this graph, , adds 2 to each value of x before squaring it. What effect does this have on the graph of It shifts the graph horizontally to the left by 2 units.
In general, if c is positive, shifts the graph of f to the left c units and shifts the graph of f to the right c units. These are called horizontal shifts of the graph of f.
Horizontal Shifts
Let f be a function and c a positive real number.
The graph of is the graph of shifted to the left c units.
The graph of is the graph of shifted to the right c units.
Some functions can be graphed by combining horizontal and vertical shifts. These functions will be variations of a function whose equation you know how to graph, such as the standard quadratic function, the standard cubic function, the square root function, the cube root function, or the absolute value function.
In our next example, we will use the graph of the standard quadratic function, , to obtain the graph of . We will graph three functions:
Objective 3: Use horizontal shifts to graph functions
Horizontal Shifts
Objective 3 Use horizontal shifts to graph functions.
Watch Video
We return to the graph of , the standard quadratic function. In Figure 1.61, the graph of function f is in the middle of the three graphs. By contrast to the vertical shift situation, this time there are graphs to the left and to the right of the graph of f. Look at the blue graph on the right. The equation of this graph, , subtracts 3 from each value of x before squaring it. What effect does this have on the graph of It shifts the graph horizontally to the right by 3 units.
Does it seem strange that subtracting 3 in the domain causes a shift of 3 units to the right? Perhaps a partial table of coordinates for each function will numerically convince you of this shift.
x
0
1
2
x
1
2
3
4
5
Notice that for the values of and g(x) to be the same, the values of x used in graphing g must each be 3 units greater than those used to graph f. For this reason, the graph of g is the graph of f shifted 3 units to the right.
Now, look at the red graph on the left in Figure 1.61. The equation of this graph, , adds 2 to each value of x before squaring it. What effect does this have on the graph of It shifts the graph horizontally to the left by 2 units.
In general, if c is positive, shifts the graph of f to the left c units and shifts the graph of f to the right c units. These are called horizontal shifts of the graph of f.
Horizontal Shifts
Let f be a function and c a positive real number.
The graph of is the graph of shifted to the left c units.
The graph of is the graph of shifted to the right c units.
Some functions can be graphed by combining horizontal and vertical shifts. These functions will be variations of a function whose equation you know how to graph, such as the standard quadratic function, the standard cubic function, the square root function, the cube root function, or the absolute value function.
In our next example, we will use the graph of the standard quadratic function, , to obtain the graph of . We will graph three functions:
This photograph shows a reflection of an old bridge in a river. This perfect reflection occurs because the surface of the water is absolutely still. A mild breeze rippling the water’s surface would distort the reflection.
Is it possible for graphs to have mirror-like qualities? Yes. Figure 1.62 shows the graphs of and . The graph of g is a reflection about the x-axis of the graph of f. For corresponding values of x, the y-coordinates of g are the opposites of the y-coordinates of f. In general, the graph of reflects the graph of f about the x-axis. Thus, the graph of g is a reflection of the graph of f about the x-axis because
Objective 5: Use vertical stretching and shrinking to graph functions
Vertical Stretching and Shrinking
Objective 5 Use vertical stretching and shrinking to graph functions.
Watch Video
Morphing does much more than move an image horizontally, vertically, or about an axis. An object having one shape is transformed into a different shape. Horizontal shifts, vertical shifts, and reflections do not change the basic shape of a graph. Graphs remain rigid and proportionally the same when they undergo these transformations. How can we shrink and stretch graphs, thereby altering their basic shapes?
Look at the three graphs in Figure 1.63. The black graph in the middle is the graph of the standard quadratic function, . Now, look at the blue graph on the top. The equation of this graph is , or . Thus, for each x, the y-coordinate of g is two times as large as the corresponding y-coordinate on the graph of f. The result is a narrower graph because the values of y are rising faster. We say that the graph of g is obtained by vertically stretching the graph of f. Now, look at the red graph on the bottom. The equation of this graph is , or . Thus, for each x, the y-coordinate of h is one-half as large as the corresponding y-coordinate on the graph of f. The result is a wider graph because the values of y are rising more slowly. We say that the graph of h is obtained by vertically shrinking the graph of f.
Objective 5: Use vertical stretching and shrinking to graph functions
Vertical Stretching and Shrinking
Objective 5 Use vertical stretching and shrinking to graph functions.
Watch Video
Morphing does much more than move an image horizontally, vertically, or about an axis. An object having one shape is transformed into a different shape. Horizontal shifts, vertical shifts, and reflections do not change the basic shape of a graph. Graphs remain rigid and proportionally the same when they undergo these transformations. How can we shrink and stretch graphs, thereby altering their basic shapes?
Look at the three graphs in Figure 1.63. The black graph in the middle is the graph of the standard quadratic function, . Now, look at the blue graph on the top. The equation of this graph is , or . Thus, for each x, the y-coordinate of g is two times as large as the corresponding y-coordinate on the graph of f. The result is a narrower graph because the values of y are rising faster. We say that the graph of g is obtained by vertically stretching the graph of f. Now, look at the red graph on the bottom. The equation of this graph is , or . Thus, for each x, the y-coordinate of h is one-half as large as the corresponding y-coordinate on the graph of f. The result is a wider graph because the values of y are rising more slowly. We say that the graph of h is obtained by vertically shrinking the graph of f.
In Exercises 1–16, use the graph of to graph each function g.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
In Exercises 17–32, use the graph of to graph each function g.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–44, use the graph of to graph each function g.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
In Exercises 45–52, use the graph of to graph each function g.
45.
46.
47.
48.
49.
50.
51.
52.
In Exercises 53–66, begin by graphing the standard quadratic function, . Then use transformations of this graph to graph the given function.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
In Exercises 67–80, begin by graphing the square root function, . Then use transformations of this graph to graph the given function.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
In Exercises 81–94, begin by graphing the absolute value function, . Then use transformations of this graph to graph the given function.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
In Exercises 95–106, begin by graphing the standard cubic function, . Then use transformations of this graph to graph the given function.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
In Exercises 107–118, begin by graphing the cube root function, . Then use transformations of this graph to graph the given function.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
Practice PLUS
In Exercises 119–122, use transformations of the graph of the greatest integer function, to graph each function. (The graph of is shown in Figure 1.39.)
119.
120.
121.
122.
In Exercises 123–126, write a possible equation for the function whose graph is shown. Each graph shows a transformation of a common function.
Describe how the graph can be obtained using transformations of the square root function .
According to the model, what is the median height of boys who are 48 months, or four years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for boys at 48 months is 40.8 inches. How well does the model describe the actual height?
Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (c)? How is this difference shown by the graph?
128. The function models the median height, , in inches, of girls who are x months of age. The graph of f is shown.
Source: Laura Walther Nathanson, The Portable Pediatrician for Parents
Describe how the graph can be obtained using transformations of the square root function .
According to the model, what is the median height of girls who are 48 months, or 4 years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for girls at 48 months is 40.2 inches. How well does the model describe the actual height?
Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (c)? How is this difference shown by the graph?
Explaining the Concepts
129. What must be done to a function’s equation so that its graph is shifted vertically upward?
130. What must be done to a function’s equation so that its graph is shifted horizontally to the right?
131. What must be done to a function’s equation so that its graph is reflected about the x-axis?
132. What must be done to a function’s equation so that its graph is reflected about the y-axis?
133. What must be done to a function’s equation so that its graph is stretched vertically?
134. What must be done to a function’s equation so that its graph is shrunk horizontally?
Technology Exercises
135.
Use a graphing utility to graph .
Graph , and in the same viewing rectangle.
Describe the relationship among the graphs of f, g, h, and k, with emphasis on different values of x for points on all four graphs that give the same y-coordinate.
Generalize by describing the relationship between the graph of f and the graph of g, where for .
Try out your generalization by sketching the graphs of for , and for a function of your choice.
136.
Use a graphing utility to graph .
Graph , and in the same viewing rectangle.
Describe the relationship among the graphs of f, g, and h, with emphasis on different values of x for points on all three graphs that give the same y-coordinate.
Generalize by describing the relationship between the graph of f and the graph of g, where for .
Try out your generalization by sketching the graphs of f(cx) for , and , and for a function of your choice.
Critical Thinking Exercises
Make Sense? During the winter, you program your home thermostat so that at midnight, the temperature is This temperature is maintained until 6 a.m. Then the house begins to warm up so that by 9 a.m. the temperature is At 6 p.m. the house begins to cool. By 9 p.m., the temperature is again The graph illustrates home temperature, , as a function of hours after midnight, t.
In Exercises 137–140, determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain [0, 24]. If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain [0, 24].
137. I decided to keep the house warmer than before, so I reprogrammed the thermostat to .
138. I decided to keep the house cooler than before, so I reprogrammed the thermostat to .
139. I decided to change the heating schedule to start one hour earlier than before, so I reprogrammed the thermostat to .
140. I decided to change the heating schedule to start one hour later than before, so I reprogrammed the thermostat to .
In Exercises 141–144, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
141. If and , then the graph of g is a translation of the graph of f 3 units to the right and 3 units upward.
142. If and , then f and g have identical graphs.
143. If and , then the graph of g can be obtained from the graph of f by stretching f 5 units followed by a downward shift of 2 units.
144. If and , then the graph of g can be obtained from the graph of f by moving f 3 units to the right, reflecting about the x-axis, and then moving the resulting graph down 4 units.
In Exercises 145–148, functions f and g are graphed in the same rectangular coordinate system. If g is obtained from f through a sequence of transformations, find an equation for g.
For Exercises 149–152, assume that (a, b) is a point on the graph of f. What is the corresponding point on the graph of each of the following functions?
149.
150.
151.
152.
Retaining the Concepts
153. The length of a rectangle exceeds the width by 13 yards. If the perimeter of the rectangle is 82 yards, what are its dimensions?
Exercises 156–158 will help you prepare for the material covered in the next section.
In Exercises 156–157, perform the indicated operation or operations.
156.
157. , where
158. Simplify:
1.6: Exercise Set
1.6 Exercise Set
Practice Exercises
In Exercises 1–16, use the graph of to graph each function g.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
In Exercises 17–32, use the graph of to graph each function g.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–44, use the graph of to graph each function g.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
In Exercises 45–52, use the graph of to graph each function g.
45.
46.
47.
48.
49.
50.
51.
52.
In Exercises 53–66, begin by graphing the standard quadratic function, . Then use transformations of this graph to graph the given function.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
In Exercises 67–80, begin by graphing the square root function, . Then use transformations of this graph to graph the given function.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
In Exercises 81–94, begin by graphing the absolute value function, . Then use transformations of this graph to graph the given function.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
In Exercises 95–106, begin by graphing the standard cubic function, . Then use transformations of this graph to graph the given function.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
In Exercises 107–118, begin by graphing the cube root function, . Then use transformations of this graph to graph the given function.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
Practice PLUS
In Exercises 119–122, use transformations of the graph of the greatest integer function, to graph each function. (The graph of is shown in Figure 1.39.)
119.
120.
121.
122.
In Exercises 123–126, write a possible equation for the function whose graph is shown. Each graph shows a transformation of a common function.
Describe how the graph can be obtained using transformations of the square root function .
According to the model, what is the median height of boys who are 48 months, or four years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for boys at 48 months is 40.8 inches. How well does the model describe the actual height?
Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (c)? How is this difference shown by the graph?
128. The function models the median height, , in inches, of girls who are x months of age. The graph of f is shown.
Source: Laura Walther Nathanson, The Portable Pediatrician for Parents
Describe how the graph can be obtained using transformations of the square root function .
According to the model, what is the median height of girls who are 48 months, or 4 years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for girls at 48 months is 40.2 inches. How well does the model describe the actual height?
Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (c)? How is this difference shown by the graph?
Explaining the Concepts
129. What must be done to a function’s equation so that its graph is shifted vertically upward?
130. What must be done to a function’s equation so that its graph is shifted horizontally to the right?
131. What must be done to a function’s equation so that its graph is reflected about the x-axis?
132. What must be done to a function’s equation so that its graph is reflected about the y-axis?
133. What must be done to a function’s equation so that its graph is stretched vertically?
134. What must be done to a function’s equation so that its graph is shrunk horizontally?
Technology Exercises
135.
Use a graphing utility to graph .
Graph , and in the same viewing rectangle.
Describe the relationship among the graphs of f, g, h, and k, with emphasis on different values of x for points on all four graphs that give the same y-coordinate.
Generalize by describing the relationship between the graph of f and the graph of g, where for .
Try out your generalization by sketching the graphs of for , and for a function of your choice.
136.
Use a graphing utility to graph .
Graph , and in the same viewing rectangle.
Describe the relationship among the graphs of f, g, and h, with emphasis on different values of x for points on all three graphs that give the same y-coordinate.
Generalize by describing the relationship between the graph of f and the graph of g, where for .
Try out your generalization by sketching the graphs of f(cx) for , and , and for a function of your choice.
Critical Thinking Exercises
Make Sense? During the winter, you program your home thermostat so that at midnight, the temperature is This temperature is maintained until 6 a.m. Then the house begins to warm up so that by 9 a.m. the temperature is At 6 p.m. the house begins to cool. By 9 p.m., the temperature is again The graph illustrates home temperature, , as a function of hours after midnight, t.
In Exercises 137–140, determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain [0, 24]. If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain [0, 24].
137. I decided to keep the house warmer than before, so I reprogrammed the thermostat to .
138. I decided to keep the house cooler than before, so I reprogrammed the thermostat to .
139. I decided to change the heating schedule to start one hour earlier than before, so I reprogrammed the thermostat to .
140. I decided to change the heating schedule to start one hour later than before, so I reprogrammed the thermostat to .
In Exercises 141–144, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
141. If and , then the graph of g is a translation of the graph of f 3 units to the right and 3 units upward.
142. If and , then f and g have identical graphs.
143. If and , then the graph of g can be obtained from the graph of f by stretching f 5 units followed by a downward shift of 2 units.
144. If and , then the graph of g can be obtained from the graph of f by moving f 3 units to the right, reflecting about the x-axis, and then moving the resulting graph down 4 units.
In Exercises 145–148, functions f and g are graphed in the same rectangular coordinate system. If g is obtained from f through a sequence of transformations, find an equation for g.
For Exercises 149–152, assume that (a, b) is a point on the graph of f. What is the corresponding point on the graph of each of the following functions?
149.
150.
151.
152.
Retaining the Concepts
153. The length of a rectangle exceeds the width by 13 yards. If the perimeter of the rectangle is 82 yards, what are its dimensions?
Exercises 156–158 will help you prepare for the material covered in the next section.
In Exercises 156–157, perform the indicated operation or operations.
156.
157. , where
158. Simplify:
1.6: Exercise Set
1.6 Exercise Set
Practice Exercises
In Exercises 1–16, use the graph of to graph each function g.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
In Exercises 17–32, use the graph of to graph each function g.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
In Exercises 33–44, use the graph of to graph each function g.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
In Exercises 45–52, use the graph of to graph each function g.
45.
46.
47.
48.
49.
50.
51.
52.
In Exercises 53–66, begin by graphing the standard quadratic function, . Then use transformations of this graph to graph the given function.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
In Exercises 67–80, begin by graphing the square root function, . Then use transformations of this graph to graph the given function.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
In Exercises 81–94, begin by graphing the absolute value function, . Then use transformations of this graph to graph the given function.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
In Exercises 95–106, begin by graphing the standard cubic function, . Then use transformations of this graph to graph the given function.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
In Exercises 107–118, begin by graphing the cube root function, . Then use transformations of this graph to graph the given function.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
Practice PLUS
In Exercises 119–122, use transformations of the graph of the greatest integer function, to graph each function. (The graph of is shown in Figure 1.39.)
119.
120.
121.
122.
In Exercises 123–126, write a possible equation for the function whose graph is shown. Each graph shows a transformation of a common function.
Describe how the graph can be obtained using transformations of the square root function .
According to the model, what is the median height of boys who are 48 months, or four years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for boys at 48 months is 40.8 inches. How well does the model describe the actual height?
Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (c)? How is this difference shown by the graph?
128. The function models the median height, , in inches, of girls who are x months of age. The graph of f is shown.
Source: Laura Walther Nathanson, The Portable Pediatrician for Parents
Describe how the graph can be obtained using transformations of the square root function .
According to the model, what is the median height of girls who are 48 months, or 4 years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for girls at 48 months is 40.2 inches. How well does the model describe the actual height?
Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (c)? How is this difference shown by the graph?
Explaining the Concepts
129. What must be done to a function’s equation so that its graph is shifted vertically upward?
130. What must be done to a function’s equation so that its graph is shifted horizontally to the right?
131. What must be done to a function’s equation so that its graph is reflected about the x-axis?
132. What must be done to a function’s equation so that its graph is reflected about the y-axis?
133. What must be done to a function’s equation so that its graph is stretched vertically?
134. What must be done to a function’s equation so that its graph is shrunk horizontally?
Technology Exercises
135.
Use a graphing utility to graph .
Graph , and in the same viewing rectangle.
Describe the relationship among the graphs of f, g, h, and k, with emphasis on different values of x for points on all four graphs that give the same y-coordinate.
Generalize by describing the relationship between the graph of f and the graph of g, where for .
Try out your generalization by sketching the graphs of for , and for a function of your choice.
136.
Use a graphing utility to graph .
Graph , and in the same viewing rectangle.
Describe the relationship among the graphs of f, g, and h, with emphasis on different values of x for points on all three graphs that give the same y-coordinate.
Generalize by describing the relationship between the graph of f and the graph of g, where for .
Try out your generalization by sketching the graphs of f(cx) for , and , and for a function of your choice.
Critical Thinking Exercises
Make Sense? During the winter, you program your home thermostat so that at midnight, the temperature is This temperature is maintained until 6 a.m. Then the house begins to warm up so that by 9 a.m. the temperature is At 6 p.m. the house begins to cool. By 9 p.m., the temperature is again The graph illustrates home temperature, , as a function of hours after midnight, t.
In Exercises 137–140, determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain [0, 24]. If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain [0, 24].
137. I decided to keep the house warmer than before, so I reprogrammed the thermostat to .
138. I decided to keep the house cooler than before, so I reprogrammed the thermostat to .
139. I decided to change the heating schedule to start one hour earlier than before, so I reprogrammed the thermostat to .
140. I decided to change the heating schedule to start one hour later than before, so I reprogrammed the thermostat to .
In Exercises 141–144, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
141. If and , then the graph of g is a translation of the graph of f 3 units to the right and 3 units upward.
142. If and , then f and g have identical graphs.
143. If and , then the graph of g can be obtained from the graph of f by stretching f 5 units followed by a downward shift of 2 units.
144. If and , then the graph of g can be obtained from the graph of f by moving f 3 units to the right, reflecting about the x-axis, and then moving the resulting graph down 4 units.
In Exercises 145–148, functions f and g are graphed in the same rectangular coordinate system. If g is obtained from f through a sequence of transformations, find an equation for g.
For Exercises 149–152, assume that (a, b) is a point on the graph of f. What is the corresponding point on the graph of each of the following functions?
149.
150.
151.
152.
Retaining the Concepts
153. The length of a rectangle exceeds the width by 13 yards. If the perimeter of the rectangle is 82 yards, what are its dimensions?
In this section, we look at these data from the perspective of functions. By considering the yearly change in the U.S. population, you will see that functions can be subtracted using procedures that will remind you of combining algebraic expressions.
Objective 1: Find the domain of a function
The Domain of a Function
Objective 1 Find the domain of a function.
Watch Video
We begin with two functions that model the data in Figure 1.66.
The data in Figure 1.66 show even-numbered years from 2000 through 2016. Because x represents the number of years after 2000,
and
Functions that model data often have their domains explicitly given with the function’s equation. However, for most functions, only an equation is given and the domain is not specified. In cases like this, the domain of a function f is the largest set of real numbers for which the value of is a real number. For example, consider the function
Because division by 0 is undefined, the denominator, , cannot be 0. Thus, x cannot equal 3. The domain of the function consists of all real numbers other than 3, represented by
Using interval notation,
Now consider a function involving a square root:
Because only nonnegative numbers have square roots that are real numbers, the expression under the square root sign, , must be nonnegative. We can use inspection to see that if . The domain of g consists of all real numbers that are greater than or equal to 3:
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in an even root, such as a square root, of a negative number.
Example 1 Finding the Domain of a Function
Find the domain of each function:
Solution
The domain is the set of all real numbers, , unless x appears in a denominator or in an even root, such as a square root.
The function contains neither division nor a square root. For every real number, x, the algebraic expression represents a real number. Thus, the domain of f is the set of all real numbers.
The function contains division. Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator, , to be 0. We can identify these values by setting equal to 0.
We must exclude and 3 from the domain of
The function contains an even root. Because only nonnegative numbers have real square roots, the quantity under the radical sign, , must be greater than or equal to 0.
The domain of h consists of all real numbers greater than or equal to .
The domain is highlighted on the x-axis in Figure 1.67.
The function contains both an even root and division. Because only nonnegative numbers have real square roots, the quantity under the radical sign, , must be greater than or equal to 0. But wait, there’s more! Because division by 0 is undefined, cannot equal 0. Thus, must be strictly greater than 0.
The domain of j consists of all real numbers less than 7.
The data in Figure 1.66 show even-numbered years from 2000 through 2016. Because x represents the number of years after 2000,
and
Functions that model data often have their domains explicitly given with the function’s equation. However, for most functions, only an equation is given and the domain is not specified. In cases like this, the domain of a function f is the largest set of real numbers for which the value of is a real number. For example, consider the function
Because division by 0 is undefined, the denominator, , cannot be 0. Thus, x cannot equal 3. The domain of the function consists of all real numbers other than 3, represented by
Using interval notation,
Now consider a function involving a square root:
Because only nonnegative numbers have square roots that are real numbers, the expression under the square root sign, , must be nonnegative. We can use inspection to see that if . The domain of g consists of all real numbers that are greater than or equal to 3:
Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in an even root, such as a square root, of a negative number.
Example 1 Finding the Domain of a Function
Find the domain of each function:
Solution
The domain is the set of all real numbers, , unless x appears in a denominator or in an even root, such as a square root.
The function contains neither division nor a square root. For every real number, x, the algebraic expression represents a real number. Thus, the domain of f is the set of all real numbers.
The function contains division. Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator, , to be 0. We can identify these values by setting equal to 0.
We must exclude and 3 from the domain of
The function contains an even root. Because only nonnegative numbers have real square roots, the quantity under the radical sign, , must be greater than or equal to 0.
The domain of h consists of all real numbers greater than or equal to .
The domain is highlighted on the x-axis in Figure 1.67.
The function contains both an even root and division. Because only nonnegative numbers have real square roots, the quantity under the radical sign, , must be greater than or equal to 0. But wait, there’s more! Because division by 0 is undefined, cannot equal 0. Thus, must be strictly greater than 0.
The domain of j consists of all real numbers less than 7.
Objective 2: Combine functions using the algebra of functions, specifying domains
The Algebra of Functions
Objective 2 Combine functions using the algebra of functions, specifying domains.
Watch Video
We can combine functions using addition, subtraction, multiplication, and division by performing operations with the algebraic expressions that appear on the right side of the equations. For example, the functions and can be combined to form the sum, difference, product, and quotient of f and g. Here’s how it’s done:
The domain for each of these functions consists of all real numbers that are common to the domains of f and g. Using to represent the domain of f and to represent the domain of g, the domain for each function is . In the case of the quotient function , we must remember not to divide by 0, so we add the further restriction that .
The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum the difference the productfg, and the quotient are functions whose domains are the set of all real numbers common to the domains of f and , defined as follows:
1. Sum:
2. Difference:
3. Product:
4. Quotient:
, provided .
Example 2 Combining Functions
Let and . Find each of the following functions:
.
Determine the domain for each function.
Solution
Because the equations for f and g do not involve division or contain even roots, the domain of both f and g is the set of all real numbers. Thus, the domain of and fg is the set of all real numbers, .
The function contains division. We must exclude from its domain values of x that cause the denominator, , to be 0. Let’s identify these values.
Example 3 Adding Functions and Determining the Domain
Let and . Find each of the following:
the domain of
Solution
The domain of is the set of all real numbers that are common to the domain of f and the domain of g. Thus, we must find the domains of f and g before finding their intersection.
Now, we can use a number line to determine , the domain of Figure 1.68 shows the domain of f, , in blue and the domain of g, , in red. Can you see that all real numbers greater than or equal to 2 are common to both domains? This is shown in purple on the number line. Thus, the domain of is .
Write a function that models the change in U.S. population, , for each year from 2000 through 2016.
Use the function from part (a) to find the change in U.S. population in 2014.
Does the result in part (b) overestimate or underestimate the actual population change in 2014 obtained from the data in Figure 1.66? By how much?
Solution
The change in population is the number of births minus the number of deaths. Thus, we will find the difference function, .
The function
models the change in U.S. population, in thousands, x years after 2000.
Because 2014 is 14 years after 2000, we substitute 14 for x in the difference function
We see that The model indicates that there was a population increase of 1357.4 thousand, or approximately 1,357,400 people, in 2014.
The data for 2014 in Figure 1.66 show 3988 thousand births and 2628 thousand deaths.
The actual population increase was 1360 thousand, or 1,360,000. Our model gave us an increase of 1357.4 thousand. Thus, the model underestimates the actual increase by , or 2.6 thousand people.
Write a function that models the total number of births and deaths in the United States for the years from 2000 through 2016.
Use the function from part (a) to find the total number of births and deaths in the United States in 2000.
Does the result in part (b) overestimate or underestimate the actual number of total births and deaths in 2000 obtained from the data in Figure 1.66? By how much?
Objective 2: Combine functions using the algebra of functions, specifying domains
The Algebra of Functions
Objective 2 Combine functions using the algebra of functions, specifying domains.
Watch Video
We can combine functions using addition, subtraction, multiplication, and division by performing operations with the algebraic expressions that appear on the right side of the equations. For example, the functions and can be combined to form the sum, difference, product, and quotient of f and g. Here’s how it’s done:
The domain for each of these functions consists of all real numbers that are common to the domains of f and g. Using to represent the domain of f and to represent the domain of g, the domain for each function is . In the case of the quotient function , we must remember not to divide by 0, so we add the further restriction that .
The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum the difference the productfg, and the quotient are functions whose domains are the set of all real numbers common to the domains of f and , defined as follows:
1. Sum:
2. Difference:
3. Product:
4. Quotient:
, provided .
Example 2 Combining Functions
Let and . Find each of the following functions:
.
Determine the domain for each function.
Solution
Because the equations for f and g do not involve division or contain even roots, the domain of both f and g is the set of all real numbers. Thus, the domain of and fg is the set of all real numbers, .
The function contains division. We must exclude from its domain values of x that cause the denominator, , to be 0. Let’s identify these values.
Example 3 Adding Functions and Determining the Domain
Let and . Find each of the following:
the domain of
Solution
The domain of is the set of all real numbers that are common to the domain of f and the domain of g. Thus, we must find the domains of f and g before finding their intersection.
Now, we can use a number line to determine , the domain of Figure 1.68 shows the domain of f, , in blue and the domain of g, , in red. Can you see that all real numbers greater than or equal to 2 are common to both domains? This is shown in purple on the number line. Thus, the domain of is .
Write a function that models the change in U.S. population, , for each year from 2000 through 2016.
Use the function from part (a) to find the change in U.S. population in 2014.
Does the result in part (b) overestimate or underestimate the actual population change in 2014 obtained from the data in Figure 1.66? By how much?
Solution
The change in population is the number of births minus the number of deaths. Thus, we will find the difference function, .
The function
models the change in U.S. population, in thousands, x years after 2000.
Because 2014 is 14 years after 2000, we substitute 14 for x in the difference function
We see that The model indicates that there was a population increase of 1357.4 thousand, or approximately 1,357,400 people, in 2014.
The data for 2014 in Figure 1.66 show 3988 thousand births and 2628 thousand deaths.
The actual population increase was 1360 thousand, or 1,360,000. Our model gave us an increase of 1357.4 thousand. Thus, the model underestimates the actual increase by , or 2.6 thousand people.
Write a function that models the total number of births and deaths in the United States for the years from 2000 through 2016.
Use the function from part (a) to find the total number of births and deaths in the United States in 2000.
Does the result in part (b) overestimate or underestimate the actual number of total births and deaths in 2000 obtained from the data in Figure 1.66? By how much?
Objective 2: Combine functions using the algebra of functions, specifying domains
The Algebra of Functions
Objective 2 Combine functions using the algebra of functions, specifying domains.
Watch Video
We can combine functions using addition, subtraction, multiplication, and division by performing operations with the algebraic expressions that appear on the right side of the equations. For example, the functions and can be combined to form the sum, difference, product, and quotient of f and g. Here’s how it’s done:
The domain for each of these functions consists of all real numbers that are common to the domains of f and g. Using to represent the domain of f and to represent the domain of g, the domain for each function is . In the case of the quotient function , we must remember not to divide by 0, so we add the further restriction that .
The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum the difference the productfg, and the quotient are functions whose domains are the set of all real numbers common to the domains of f and , defined as follows:
1. Sum:
2. Difference:
3. Product:
4. Quotient:
, provided .
Example 2 Combining Functions
Let and . Find each of the following functions:
.
Determine the domain for each function.
Solution
Because the equations for f and g do not involve division or contain even roots, the domain of both f and g is the set of all real numbers. Thus, the domain of and fg is the set of all real numbers, .
The function contains division. We must exclude from its domain values of x that cause the denominator, , to be 0. Let’s identify these values.
Example 3 Adding Functions and Determining the Domain
Let and . Find each of the following:
the domain of
Solution
The domain of is the set of all real numbers that are common to the domain of f and the domain of g. Thus, we must find the domains of f and g before finding their intersection.
Now, we can use a number line to determine , the domain of Figure 1.68 shows the domain of f, , in blue and the domain of g, , in red. Can you see that all real numbers greater than or equal to 2 are common to both domains? This is shown in purple on the number line. Thus, the domain of is .
Write a function that models the change in U.S. population, , for each year from 2000 through 2016.
Use the function from part (a) to find the change in U.S. population in 2014.
Does the result in part (b) overestimate or underestimate the actual population change in 2014 obtained from the data in Figure 1.66? By how much?
Solution
The change in population is the number of births minus the number of deaths. Thus, we will find the difference function, .
The function
models the change in U.S. population, in thousands, x years after 2000.
Because 2014 is 14 years after 2000, we substitute 14 for x in the difference function
We see that The model indicates that there was a population increase of 1357.4 thousand, or approximately 1,357,400 people, in 2014.
The data for 2014 in Figure 1.66 show 3988 thousand births and 2628 thousand deaths.
The actual population increase was 1360 thousand, or 1,360,000. Our model gave us an increase of 1357.4 thousand. Thus, the model underestimates the actual increase by , or 2.6 thousand people.
Write a function that models the total number of births and deaths in the United States for the years from 2000 through 2016.
Use the function from part (a) to find the total number of births and deaths in the United States in 2000.
Does the result in part (b) overestimate or underestimate the actual number of total births and deaths in 2000 obtained from the data in Figure 1.66? By how much?
Objective 2: Combine functions using the algebra of functions, specifying domains
The Algebra of Functions
Objective 2 Combine functions using the algebra of functions, specifying domains.
Watch Video
We can combine functions using addition, subtraction, multiplication, and division by performing operations with the algebraic expressions that appear on the right side of the equations. For example, the functions and can be combined to form the sum, difference, product, and quotient of f and g. Here’s how it’s done:
The domain for each of these functions consists of all real numbers that are common to the domains of f and g. Using to represent the domain of f and to represent the domain of g, the domain for each function is . In the case of the quotient function , we must remember not to divide by 0, so we add the further restriction that .
The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum the difference the productfg, and the quotient are functions whose domains are the set of all real numbers common to the domains of f and , defined as follows:
1. Sum:
2. Difference:
3. Product:
4. Quotient:
, provided .
Example 2 Combining Functions
Let and . Find each of the following functions:
.
Determine the domain for each function.
Solution
Because the equations for f and g do not involve division or contain even roots, the domain of both f and g is the set of all real numbers. Thus, the domain of and fg is the set of all real numbers, .
The function contains division. We must exclude from its domain values of x that cause the denominator, , to be 0. Let’s identify these values.
Example 3 Adding Functions and Determining the Domain
Let and . Find each of the following:
the domain of
Solution
The domain of is the set of all real numbers that are common to the domain of f and the domain of g. Thus, we must find the domains of f and g before finding their intersection.
Now, we can use a number line to determine , the domain of Figure 1.68 shows the domain of f, , in blue and the domain of g, , in red. Can you see that all real numbers greater than or equal to 2 are common to both domains? This is shown in purple on the number line. Thus, the domain of is .
Write a function that models the change in U.S. population, , for each year from 2000 through 2016.
Use the function from part (a) to find the change in U.S. population in 2014.
Does the result in part (b) overestimate or underestimate the actual population change in 2014 obtained from the data in Figure 1.66? By how much?
Solution
The change in population is the number of births minus the number of deaths. Thus, we will find the difference function, .
The function
models the change in U.S. population, in thousands, x years after 2000.
Because 2014 is 14 years after 2000, we substitute 14 for x in the difference function
We see that The model indicates that there was a population increase of 1357.4 thousand, or approximately 1,357,400 people, in 2014.
The data for 2014 in Figure 1.66 show 3988 thousand births and 2628 thousand deaths.
The actual population increase was 1360 thousand, or 1,360,000. Our model gave us an increase of 1357.4 thousand. Thus, the model underestimates the actual increase by , or 2.6 thousand people.
Write a function that models the total number of births and deaths in the United States for the years from 2000 through 2016.
Use the function from part (a) to find the total number of births and deaths in the United States in 2000.
Does the result in part (b) overestimate or underestimate the actual number of total births and deaths in 2000 obtained from the data in Figure 1.66? By how much?
Objective 3: Form composite functions
Composite Functions
Objective 3 Form composite functions.
Watch Video
There is another way of combining two functions. To help understand this new combination, suppose that your local computer store is having a sale. The models that are on sale cost either $300 less than the regular price or 85% of the regular price. If x represents the computer’s regular price, the discounts can be modeled with the following functions:
At the store, you bargain with the salesperson. Eventually, she makes an offer you can’t refuse. The sale price will be 85% of the regular price followed by a $300 reduction:
In terms of the functions f and g, this offer can be obtained by taking the output of , namely, 0.85x, and using it as the input of f:
Because 0.85x is g(x), we can write this last equation as
We read this equation as “f of g of x is equal to .” We call the composition of the function f with g, or a composite function. This composite function is written Thus,
Like all functions, we can evaluate for a specified value of x in the function’s domain. For example, here’s how to find the value of the composite function describing the offer you cannot refuse at 1400:
This means that a computer that regularly sells for $1400 is on sale for $890 subject to both discounts. We can use a partial table of coordinates for each of the discount functions, g and f, to verify this result numerically.
This verifies that a computer that regularly sells for $1400 is on sale for $890 subject to both discounts.
Before you run out to buy a computer, let’s generalize our discussion of the computer’s double discount and define the composition of any two functions.
The Composition of Functions
The composition of the function f with g is denoted by and is defined by the equation
The domain of the composite function is the set of all x such that
x is in the domain of g and
g(x) is in the domain of f.
The composition of f with is illustrated in Figure 1.69.
There is another way of combining two functions. To help understand this new combination, suppose that your local computer store is having a sale. The models that are on sale cost either $300 less than the regular price or 85% of the regular price. If x represents the computer’s regular price, the discounts can be modeled with the following functions:
At the store, you bargain with the salesperson. Eventually, she makes an offer you can’t refuse. The sale price will be 85% of the regular price followed by a $300 reduction:
In terms of the functions f and g, this offer can be obtained by taking the output of , namely, 0.85x, and using it as the input of f:
Because 0.85x is g(x), we can write this last equation as
We read this equation as “f of g of x is equal to .” We call the composition of the function f with g, or a composite function. This composite function is written Thus,
Like all functions, we can evaluate for a specified value of x in the function’s domain. For example, here’s how to find the value of the composite function describing the offer you cannot refuse at 1400:
This means that a computer that regularly sells for $1400 is on sale for $890 subject to both discounts. We can use a partial table of coordinates for each of the discount functions, g and f, to verify this result numerically.
This verifies that a computer that regularly sells for $1400 is on sale for $890 subject to both discounts.
Before you run out to buy a computer, let’s generalize our discussion of the computer’s double discount and define the composition of any two functions.
The Composition of Functions
The composition of the function f with g is denoted by and is defined by the equation
The domain of the composite function is the set of all x such that
x is in the domain of g and
g(x) is in the domain of f.
The composition of f with is illustrated in Figure 1.69.
There is another way of combining two functions. To help understand this new combination, suppose that your local computer store is having a sale. The models that are on sale cost either $300 less than the regular price or 85% of the regular price. If x represents the computer’s regular price, the discounts can be modeled with the following functions:
At the store, you bargain with the salesperson. Eventually, she makes an offer you can’t refuse. The sale price will be 85% of the regular price followed by a $300 reduction:
In terms of the functions f and g, this offer can be obtained by taking the output of , namely, 0.85x, and using it as the input of f:
Because 0.85x is g(x), we can write this last equation as
We read this equation as “f of g of x is equal to .” We call the composition of the function f with g, or a composite function. This composite function is written Thus,
Like all functions, we can evaluate for a specified value of x in the function’s domain. For example, here’s how to find the value of the composite function describing the offer you cannot refuse at 1400:
This means that a computer that regularly sells for $1400 is on sale for $890 subject to both discounts. We can use a partial table of coordinates for each of the discount functions, g and f, to verify this result numerically.
This verifies that a computer that regularly sells for $1400 is on sale for $890 subject to both discounts.
Before you run out to buy a computer, let’s generalize our discussion of the computer’s double discount and define the composition of any two functions.
The Composition of Functions
The composition of the function f with g is denoted by and is defined by the equation
The domain of the composite function is the set of all x such that
x is in the domain of g and
g(x) is in the domain of f.
The composition of f with is illustrated in Figure 1.69.
When you form a composite function, you “compose” two functions to form a new function. It is also possible to reverse this process. That is, you can “decompose” a given function and express it as a composition of two functions. Although there is more than one way to do this, there is often a “natural” selection that comes to mind first. For example, consider the function h defined by
The function h takes and raises it to the power 5. A natural way to write h as a composition of two functions is to raise the function to the power 5. Thus, if we let
Example 7 Writing a Function as a Composition
Express h(x) as a composition of two functions:
Solution
The function h takes and takes its cube root. A natural way to write h as a composition of two functions is to take the cube root of the function . Thus, we let
We can check this composition by finding . This should give the original function, namely, .
Write a function d that models the difference between the projected population under 45 and the projected population 45 and older for the years shown in the bar graph.
Use the function from part (a) to find how many more people under 45 than 45 and older there are projected to be in 2060.
Does the result in part (b) overestimate, underestimate, or give the actual difference between the under-45 and 45-and-older populations in 2060 shown by the bar graph?
98.
Write a function r that models the ratio of the projected population 45 and older to the projected population under 45 for the years shown in the bar graph.
Use the function from part (a) to find the ratio of the projected population 45 and older to the projected population under 45, correct to two decimal places, for 2040.
Find the ratio of the projected population 45 and older to the projected population under 45, correct to two decimal places, shown by the bar graph. How does the rounded ratio in part (b) compare with this ratio?
99. A company that sells protective tablet cases has yearly fixed costs of $600,000. It costs the company $45 to produce each case. Each case will sell for $65. The company’s costs and revenue are modeled by the following functions, where x represents the number of tablet cases produced and sold:
Find and interpret and
100. A department store has two locations in a city. From 2016 through 2020, the profits for each of the store’s two branches are modeled by the functions and In each model, x represents the number of years after 2016, and f and g represent the profit, in millions of dollars.
What is the slope of f? Describe what this means.
What is the slope of g? Describe what this means.
Find What is the slope of this function? What does this mean?
101. The regular price of a laptop is x dollars. Let and
Describe what the functions f and g model in terms of the price of the laptop.
Find and describe what this models in terms of the price of the laptop.
Repeat part (b) for
Which composite function models the greater discount on the laptop, or Explain.
102. The regular price of a pair of jeans is x dollars. Let and
Describe what functions f and g model in terms of the price of the jeans.
Find and describe what this models in terms of the price of the jeans.
Repeat part (b) for
Which composite function models the greater discount on the jeans, or Explain.
Explaining the Concepts
103. If a function is defined by an equation, explain how to find its domain.
104. If equations for f and g are given, explain how to find
105. If equations for two functions are given, explain how to obtain the quotient function and its domain.
106. Describe a procedure for finding What is the name of this function?
107. Describe the values of x that must be excluded from the domain of
Technology Exercises
108. Graph and in the same by viewing rectangle. Then use the feature to trace along What happens at Explain why this occurs.
109. Graph and in the same by [0, 2, 1] viewing rectangle. If represents f and represents g, use the graph of to find the domain of Then verify your observation algebraically.
Critical Thinking Exercises
Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning.
110. I used a function to model data from 1990 through 2020. The independent variable in my model represented the number of years after 1990, so the function’s domain was
111. I have two functions. Function f models total world population x years after 2000 and function g models population of the world’s more-developed regions x years after 2000. I can use to determine the population of the world’s less-developed regions for the years in both function’s domains.
112. I must have made a mistake in finding the composite functions and because I notice that is not the same function as
113. This diagram illustrates that
In Exercises 114–117, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
114. If and then and
115. There can never be two functions f and g, where for which
116. If and then
117. If and then
118. Prove that if f and g are even functions, then fg is also an even function.
121. In July 2020, the toll for the Golden Gate Bridge was $8.40 for drivers making one-time payments for each crossing. Drivers who pay a one-time fee of $20.00 to purchase a FasTrak toll tag pay $7.70 for each crossing. How many times must a driver cross the Golden Gate Bridge for the cost of these two options to be the same? Round to the nearest whole number. Find the total cost of each option for the rounded number of crossings.
Write a function d that models the difference between the projected population under 45 and the projected population 45 and older for the years shown in the bar graph.
Use the function from part (a) to find how many more people under 45 than 45 and older there are projected to be in 2060.
Does the result in part (b) overestimate, underestimate, or give the actual difference between the under-45 and 45-and-older populations in 2060 shown by the bar graph?
98.
Write a function r that models the ratio of the projected population 45 and older to the projected population under 45 for the years shown in the bar graph.
Use the function from part (a) to find the ratio of the projected population 45 and older to the projected population under 45, correct to two decimal places, for 2040.
Find the ratio of the projected population 45 and older to the projected population under 45, correct to two decimal places, shown by the bar graph. How does the rounded ratio in part (b) compare with this ratio?
99. A company that sells protective tablet cases has yearly fixed costs of $600,000. It costs the company $45 to produce each case. Each case will sell for $65. The company’s costs and revenue are modeled by the following functions, where x represents the number of tablet cases produced and sold:
Find and interpret and
100. A department store has two locations in a city. From 2016 through 2020, the profits for each of the store’s two branches are modeled by the functions and In each model, x represents the number of years after 2016, and f and g represent the profit, in millions of dollars.
What is the slope of f? Describe what this means.
What is the slope of g? Describe what this means.
Find What is the slope of this function? What does this mean?
101. The regular price of a laptop is x dollars. Let and
Describe what the functions f and g model in terms of the price of the laptop.
Find and describe what this models in terms of the price of the laptop.
Repeat part (b) for
Which composite function models the greater discount on the laptop, or Explain.
102. The regular price of a pair of jeans is x dollars. Let and
Describe what functions f and g model in terms of the price of the jeans.
Find and describe what this models in terms of the price of the jeans.
Repeat part (b) for
Which composite function models the greater discount on the jeans, or Explain.
Explaining the Concepts
103. If a function is defined by an equation, explain how to find its domain.
104. If equations for f and g are given, explain how to find
105. If equations for two functions are given, explain how to obtain the quotient function and its domain.
106. Describe a procedure for finding What is the name of this function?
107. Describe the values of x that must be excluded from the domain of
Technology Exercises
108. Graph and in the same by viewing rectangle. Then use the feature to trace along What happens at Explain why this occurs.
109. Graph and in the same by [0, 2, 1] viewing rectangle. If represents f and represents g, use the graph of to find the domain of Then verify your observation algebraically.
Critical Thinking Exercises
Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning.
110. I used a function to model data from 1990 through 2020. The independent variable in my model represented the number of years after 1990, so the function’s domain was
111. I have two functions. Function f models total world population x years after 2000 and function g models population of the world’s more-developed regions x years after 2000. I can use to determine the population of the world’s less-developed regions for the years in both function’s domains.
112. I must have made a mistake in finding the composite functions and because I notice that is not the same function as
113. This diagram illustrates that
In Exercises 114–117, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
114. If and then and
115. There can never be two functions f and g, where for which
116. If and then
117. If and then
118. Prove that if f and g are even functions, then fg is also an even function.
121. In July 2020, the toll for the Golden Gate Bridge was $8.40 for drivers making one-time payments for each crossing. Drivers who pay a one-time fee of $20.00 to purchase a FasTrak toll tag pay $7.70 for each crossing. How many times must a driver cross the Golden Gate Bridge for the cost of these two options to be the same? Round to the nearest whole number. Find the total cost of each option for the rounded number of crossings.
Write a function d that models the difference between the projected population under 45 and the projected population 45 and older for the years shown in the bar graph.
Use the function from part (a) to find how many more people under 45 than 45 and older there are projected to be in 2060.
Does the result in part (b) overestimate, underestimate, or give the actual difference between the under-45 and 45-and-older populations in 2060 shown by the bar graph?
98.
Write a function r that models the ratio of the projected population 45 and older to the projected population under 45 for the years shown in the bar graph.
Use the function from part (a) to find the ratio of the projected population 45 and older to the projected population under 45, correct to two decimal places, for 2040.
Find the ratio of the projected population 45 and older to the projected population under 45, correct to two decimal places, shown by the bar graph. How does the rounded ratio in part (b) compare with this ratio?
99. A company that sells protective tablet cases has yearly fixed costs of $600,000. It costs the company $45 to produce each case. Each case will sell for $65. The company’s costs and revenue are modeled by the following functions, where x represents the number of tablet cases produced and sold:
Find and interpret and
100. A department store has two locations in a city. From 2016 through 2020, the profits for each of the store’s two branches are modeled by the functions and In each model, x represents the number of years after 2016, and f and g represent the profit, in millions of dollars.
What is the slope of f? Describe what this means.
What is the slope of g? Describe what this means.
Find What is the slope of this function? What does this mean?
101. The regular price of a laptop is x dollars. Let and
Describe what the functions f and g model in terms of the price of the laptop.
Find and describe what this models in terms of the price of the laptop.
Repeat part (b) for
Which composite function models the greater discount on the laptop, or Explain.
102. The regular price of a pair of jeans is x dollars. Let and
Describe what functions f and g model in terms of the price of the jeans.
Find and describe what this models in terms of the price of the jeans.
Repeat part (b) for
Which composite function models the greater discount on the jeans, or Explain.
Explaining the Concepts
103. If a function is defined by an equation, explain how to find its domain.
104. If equations for f and g are given, explain how to find
105. If equations for two functions are given, explain how to obtain the quotient function and its domain.
106. Describe a procedure for finding What is the name of this function?
107. Describe the values of x that must be excluded from the domain of
Technology Exercises
108. Graph and in the same by viewing rectangle. Then use the feature to trace along What happens at Explain why this occurs.
109. Graph and in the same by [0, 2, 1] viewing rectangle. If represents f and represents g, use the graph of to find the domain of Then verify your observation algebraically.
Critical Thinking Exercises
Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning.
110. I used a function to model data from 1990 through 2020. The independent variable in my model represented the number of years after 1990, so the function’s domain was
111. I have two functions. Function f models total world population x years after 2000 and function g models population of the world’s more-developed regions x years after 2000. I can use to determine the population of the world’s less-developed regions for the years in both function’s domains.
112. I must have made a mistake in finding the composite functions and because I notice that is not the same function as
113. This diagram illustrates that
In Exercises 114–117, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
114. If and then and
115. There can never be two functions f and g, where for which
116. If and then
117. If and then
118. Prove that if f and g are even functions, then fg is also an even function.
121. In July 2020, the toll for the Golden Gate Bridge was $8.40 for drivers making one-time payments for each crossing. Drivers who pay a one-time fee of $20.00 to purchase a FasTrak toll tag pay $7.70 for each crossing. How many times must a driver cross the Golden Gate Bridge for the cost of these two options to be the same? Round to the nearest whole number. Find the total cost of each option for the rounded number of crossings.
Based on Shakespeare’s Romeo and Juliet, the film West Side Story swept the 1961 Academy Awards with ten Oscars. The top four movies to win the most Oscars are shown in Table 1.6.
Table 1.6 Films Winning the Most Oscars
Movie
Year
Number of Academy Awards
Ben-Hur
1960
11
Titanic
1998
11
The Lord of the Rings: The Return of the King
2003
11
West Side Story
1961
10
Source: Russell Ash, The Top 10 of Everything, 2011
We can use the information in Table 1.6 to define a function. Let the domain of the function be the set of four movies shown in the table. Let the range be the number of Academy Awards for each of the respective films. The function can be written as follows:
Now let’s “undo” f by interchanging the first and second components in each of the ordered pairs. Switching the inputs and outputs of f, we obtain the following relation:
Can you see that this relation is not a function? Three of its ordered pairs have the same first component and different second components. This violates the definition of a function.
If a function f is a set of ordered pairs, then the changes produced by f can be “undone” by reversing the components of all the ordered pairs. The resulting relation, , may or may not be a function. In this section, we will develop these ideas by studying functions whose compositions have a special “undoing” relationship.
Inverse Functions
Here are two functions that describe situations related to the price of a computer, x:
Function f subtracts $300 from the computer’s price and function g adds $300 to the computer’s price. Let’s see what does. Put g(x) into f:
Using and , we see that . By putting g(x) into f and finding , the computer’s price, x, went through two changes: the first, an increase; the second, a decrease:
The final price of the computer, x, is identical to its starting price, x.
In general, if the changes made to x by a function g are undone by the changes made by a function f, then
Assume, also, that this “undoing” takes place in the other direction:
Under these conditions, we say that each function is the inverse function of the other. The fact that g is the inverse of f is expressed by renaming g as , read “f-inverse.” For example, the inverse functions
are usually named as follows:
We can use partial tables of coordinates for f and to gain numerical insight into the relationship between a function and its inverse function.
The tables illustrate that if a function f is the set of ordered pairs , then its inverse, , is the set of ordered pairs . Using these tables, we can see how one function’s changes to x are undone by the other function:
The final price of the computer, $1300, is identical to its starting price, $1300.
With these ideas in mind, we present the formal definition of the inverse of a function:
Definition of the Inverse of a Function
Let f and g be two functions such that
and
The function g is the inverse of the function f and is denoted by (read “f-inverse”). Thus, and . The domain of f is equal to the range of , and vice versa.
Objective 1: Verify inverse functions
Objective 1 Verify inverse functions.
Watch Video
Example 1 Verifying Inverse Functions
Show that each function is the inverse of the other:
Solution
To show that f and g are inverses of each other, we must show that and . We begin with .
Because g is the inverse of f (and vice versa), we can use inverse notation and write
Notice how undoes the changes produced by f: f changes x by multiplying by 3 and adding 2, and undoes this by subtracting 2 and dividing by 3. This “undoing” process is illustrated in Figure 1.70.
Based on Shakespeare’s Romeo and Juliet, the film West Side Story swept the 1961 Academy Awards with ten Oscars. The top four movies to win the most Oscars are shown in Table 1.6.
Table 1.6 Films Winning the Most Oscars
Movie
Year
Number of Academy Awards
Ben-Hur
1960
11
Titanic
1998
11
The Lord of the Rings: The Return of the King
2003
11
West Side Story
1961
10
Source: Russell Ash, The Top 10 of Everything, 2011
We can use the information in Table 1.6 to define a function. Let the domain of the function be the set of four movies shown in the table. Let the range be the number of Academy Awards for each of the respective films. The function can be written as follows:
Now let’s “undo” f by interchanging the first and second components in each of the ordered pairs. Switching the inputs and outputs of f, we obtain the following relation:
Can you see that this relation is not a function? Three of its ordered pairs have the same first component and different second components. This violates the definition of a function.
If a function f is a set of ordered pairs, then the changes produced by f can be “undone” by reversing the components of all the ordered pairs. The resulting relation, , may or may not be a function. In this section, we will develop these ideas by studying functions whose compositions have a special “undoing” relationship.
Inverse Functions
Here are two functions that describe situations related to the price of a computer, x:
Function f subtracts $300 from the computer’s price and function g adds $300 to the computer’s price. Let’s see what does. Put g(x) into f:
Using and , we see that . By putting g(x) into f and finding , the computer’s price, x, went through two changes: the first, an increase; the second, a decrease:
The final price of the computer, x, is identical to its starting price, x.
In general, if the changes made to x by a function g are undone by the changes made by a function f, then
Assume, also, that this “undoing” takes place in the other direction:
Under these conditions, we say that each function is the inverse function of the other. The fact that g is the inverse of f is expressed by renaming g as , read “f-inverse.” For example, the inverse functions
are usually named as follows:
We can use partial tables of coordinates for f and to gain numerical insight into the relationship between a function and its inverse function.
The tables illustrate that if a function f is the set of ordered pairs , then its inverse, , is the set of ordered pairs . Using these tables, we can see how one function’s changes to x are undone by the other function:
The final price of the computer, $1300, is identical to its starting price, $1300.
With these ideas in mind, we present the formal definition of the inverse of a function:
Definition of the Inverse of a Function
Let f and g be two functions such that
and
The function g is the inverse of the function f and is denoted by (read “f-inverse”). Thus, and . The domain of f is equal to the range of , and vice versa.
Objective 2: Find the inverse of a function
Finding the Inverse of a Function
Objective 2 Find the inverse of a function.
Watch Video
The definition of the inverse of a function tells us that the domain of f is equal to the range of , and vice versa. This means that if the function f is the set of ordered pairs , then the inverse of f is the set of ordered pairs . If a function is defined by an equation, we can obtain the equation for the inverse of f, by interchanging the role of x and y in the equation for the function f.
Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows:
Replace with y in the equation for .
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
If f has an inverse function, replace y in step 3 by . We can verify our result by showing that and .
The procedure for finding a function’s inverse uses a switch-and-solve strategy. Switch x and y, and then solve for y.
Example 2 Finding the Inverse of a Function
Find the inverse of .
Solution
Step 1 REPLACE WITH y:
Step 2 INTERCHANGE x AND y:
Step 3 SOLVE FOR y:
Step 4 REPLACE y WITH
Thus, the inverse of is .
The inverse function, , undoes the changes produced by changes x by multiplying by 7 and subtracting 5. undoes this by adding 5 and dividing by 7.
The definition of the inverse of a function tells us that the domain of f is equal to the range of , and vice versa. This means that if the function f is the set of ordered pairs , then the inverse of f is the set of ordered pairs . If a function is defined by an equation, we can obtain the equation for the inverse of f, by interchanging the role of x and y in the equation for the function f.
Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows:
Replace with y in the equation for .
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
If f has an inverse function, replace y in step 3 by . We can verify our result by showing that and .
The procedure for finding a function’s inverse uses a switch-and-solve strategy. Switch x and y, and then solve for y.
Example 2 Finding the Inverse of a Function
Find the inverse of .
Solution
Step 1 REPLACE WITH y:
Step 2 INTERCHANGE x AND y:
Step 3 SOLVE FOR y:
Step 4 REPLACE y WITH
Thus, the inverse of is .
The inverse function, , undoes the changes produced by changes x by multiplying by 7 and subtracting 5. undoes this by adding 5 and dividing by 7.
The definition of the inverse of a function tells us that the domain of f is equal to the range of , and vice versa. This means that if the function f is the set of ordered pairs , then the inverse of f is the set of ordered pairs . If a function is defined by an equation, we can obtain the equation for the inverse of f, by interchanging the role of x and y in the equation for the function f.
Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows:
Replace with y in the equation for .
Interchange x and y.
Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
If f has an inverse function, replace y in step 3 by . We can verify our result by showing that and .
The procedure for finding a function’s inverse uses a switch-and-solve strategy. Switch x and y, and then solve for y.
Example 2 Finding the Inverse of a Function
Find the inverse of .
Solution
Step 1 REPLACE WITH y:
Step 2 INTERCHANGE x AND y:
Step 3 SOLVE FOR y:
Step 4 REPLACE y WITH
Thus, the inverse of is .
The inverse function, , undoes the changes produced by changes x by multiplying by 7 and subtracting 5. undoes this by adding 5 and dividing by 7.
Objective 3: Use the horizontal line test to determine if a function has an inverse function
The Horizontal Line Test and One-to-One Functions
Objective 3 Use the horizontal line test to determine if a function has an inverse function.
Watch Video
Let’s see what happens if we try to find the inverse of the standard quadratic function,
Step 1 Replace withy:
Step 2 Interchangexandy:
Step 3 Solve fory: We apply the square root property to solve for y. We obtain
The ± in shows that for certain values of x (all positive real numbers), there are two values of y. Because this equation does not represent y as a function of x, the standard quadratic function does not have an inverse function.
We can use a few of the solutions of to illustrate numerically that this function does not have an inverse:
A function provides exactly one output for each input. Thus, the ordered pairs in the bottom row do not define a function.
Can we look at the graph of a function and tell if it represents a function with an inverse? Yes. The graph of the standard quadratic function is shown in Figure 1.71. Four units above the x-axis, a horizontal line is drawn. This line intersects the graph at two of its points, and (2, 4). Inverse functions have ordered pairs with the coordinates reversed. We just saw what happened when we interchanged x and y. We obtained and (4, 2), and these ordered pairs do not define a function.
Figure 1.71 The horizontal line intersects the graph twice.
If any horizontal line, such as the one in Figure 1.71, intersects a graph at two or more points, the set of these points will not define a function when their coordinates are reversed. This suggests the horizontal line test for inverse functions.
The Horizontal Line Test for Inverse Functions
A function f has an inverse that is a function, , if there is no horizontal line that intersects the graph of the function f at more than one point.
Example 5 Applying the Horizontal Line Test
Which of the following graphs represent functions that have inverse functions?
Notice that horizontal lines can be drawn in graphs (b) and (c) that intersect the graphs more than once. These graphs do not pass the horizontal line test. These are not the graphs of functions with inverse functions. By contrast, no horizontal line can be drawn in graphs (a) and (d) that intersects the graphs more than once. These graphs pass the horizontal line test. Thus, the graphs in parts (a) and (d) represent functions that have inverse functions.
A function passes the horizontal line test when no two different ordered pairs have the same second component. This means that if , then . Such a function is called a one-to-one function. Thus, a one-to-one function is a function in which no two different ordered pairs have the same second component. Only one-to-one functions have inverse functions. Any function that passes the horizontal line test is a one-to-one function. Any one-to-one function has a graph that passes the horizontal line test.
Objective 4: Use the graph of a one-to-one function to graph its inverse function
Graphs of f and
Objective 4 Use the graph of a one-to-one function to graph its inverse function.
Watch Video
There is a relationship between the graph of a one-to-one function, f, and its inverse, . Because inverse functions have ordered pairs with the coordinates interchanged, if the point is on the graph of f, then the point is on the graph of . The points and are symmetric with respect to the line . Thus, the graph ofis a reflection of the graph of f about the line . This is illustrated in Figure 1.72.
Figure 1.72 The graph of is a reflection of the graph of f about .
Use the graph of f in Figure 1.73 to draw the graph of its inverse function.
Figure 1.73
Solution
We begin by noting that no horizontal line intersects the graph of f at more than one point, so f does have an inverse function. Because the points , and (4, 2) are on the graph of f, the graph of the inverse function, , has points with these ordered pairs reversed. Thus, , and (2, 4) are on the graph of . We can use these points to graph . The graph of is shown in green in Figure 1.74. Note that the green graph of is the reflection of the blue graph of f about the line .
The graph of function f consists of two line segments, one segment from to and a second segment from to (1, 2). Graph f and use the graph to draw the graph of its inverse function.
Objective 5: Find the inverse of a function and graph both functions on the same axes
Objective 5 Find the inverse of a function and graph both functions on the same axes.
Watch Video
In our final example, we will first find . Then we will graph f and in the same rectangular coordinate system.
Example 7 Finding the Inverse of a Domain-Restricted Function
Find the inverse of if . Graph f and in the same rectangular coordinate system.
Solution
The graph of is the graph of the standard quadratic function shifted vertically down 1 unit. Figure 1.75 shows the function’s graph. This graph fails the horizontal line test, so the function does not have an inverse function. By restricting the domain to , as given, we obtain a new function whose graph is shown in red in Figure 1.75. This red portion of the graph is increasing on the interval and passes the horizontal line test. This tells us that has an inverse function if we restrict its domain to . We use our four-step procedure to find this inverse function. Begin with .
Thus, the inverse of , is . The graphs of f and are shown in Figure 1.76. We obtained the graph of by shifting the graph of the square root function, , horizontally to the left 1 unit. Note that the green graph of is the reflection of the red graph of f about the line .
Find the inverse of if . Graph f and in the same rectangular coordinate system.
1.8: Exercise Set
1.8 Exercise Set
Practice Exercises
In Exercises 1–10, find and and determine whether each pair of functions f and g are inverses of each other.
1. and
2. and
3. and
4. and
5. and
6. and
7. and
8. and
9. and
10. and
The functions in Exercises 11–28 are all one-to-one. For each function,
Find an equation for , the inverse function.
Verify that your equation is correct by showing that and .
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
Which graphs in Exercises 29–34 represent functions that have inverse functions?
29.
30.
31.
32.
33.
34.
In Exercises 35–38, use the graph of f to draw the graph of its inverse function.
35.
36.
37.
38.
In Exercises 39–52,
Find an equation for .
Graph f and in the same rectangular coordinate system.
Use interval notation to give the domain and the range of f and .
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
(Hint for Exercises 49–52: To solve for a variable involving an nth root, raise both sides of the equation to the nth power: .)
49.
50.
51.
52.
Practice PLUS
In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function.
x
1
0
4
1
5
2
x
g(x)
0
1
1
4
2
10
53.
54.
55.
56.
57.
58.
In Exercises 59–64, let
Evaluate the indicated function without finding an equation for the function.
59.
60.
61.
62.
63.
64.
Application Exercises
Way to Go Holland was the first country to establish an official bicycle policy. It currently has over 12,000 miles of paths and lanes exclusively for bicycles. The graph shows the percentage of travel by bike and by car in Holland, as well as in four other selected countries. Use the information in the graph to solve Exercises 65–66.
Consider a function, f, whose domain is the set of the five countries shown in the graph. Let the range be the percentage of travel by bike in each of the respective countries. Write the function f as a set of ordered pairs.
Write the relation that is the inverse of f as a set of ordered pairs. Is this relation a function? Explain your answer.
66.
Consider a function, f, whose domain is the set of the five countries shown in the graph. Let the range be the percentage of travel by car in each of the respective countries. Write the function f as a set of ordered pairs.
Write the relation that is the inverse of f as a set of ordered pairs. Is this relation a function? Explain your answer.
67. The graph represents the probability of two people in the same room sharing a birthday as a function of the number of people in the room. Call the function f.
Describe in practical terms the meaning of , and .
68. A study of 900 working women in Texas showed that their feelings changed throughout the day. As the graph indicates, the women felt better as time passed, except for a blip (that’s slang for relative maximum) at lunchtime.
Source: D. Kahneman et al., “A Survey Method for Characterizing Daily Life Experience,” Science
Does the graph have an inverse that is a function? Explain your answer.
Identify two or more times of day when the average happiness level is 3. Express your answers as ordered pairs.
Do the ordered pairs in part (b) indicate that the graph represents a one-to-one function? Explain your answer.
69. The formula
is used to convert from x degrees Celsius to y degrees Fahrenheit. The formula
is used to convert from x degrees Fahrenheit to y degrees Celsius. Show that f and g are inverse functions.
Explaining the Concepts
70. Explain how to determine if two functions are inverses of each other.
71. Describe how to find the inverse of a one-to-one function.
72. What is the horizontal line test and what does it indicate?
73. Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.
74. How can a graphing utility be used to visually determine if two functions are inverses of each other?
75. What explanations can you offer for the trends shown by the graph in Exercise 68?
Technology Exercises
In Exercises 76–83, use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one).
76.
77.
78.
79.
80.
81.
82.
83.
In Exercises 84–86, use a graphing utility to graph f and g in the same by viewing rectangle. In addition, graph the line and visually determine if f and g are inverses.
84.
85.
86.
Critical Thinking Exercises
Make Sense? In Exercises 87–90, determine whether each statement makes sense or does not make sense, and explain your reasoning.
87. I found the inverse of in my head: The reverse of multiplying by 5 and subtracting 4 is adding 4 and dividing by 5, so .
88. I’m working with the linear function and I do not need to find in order to determine the value of .
89. When finding the inverse of a function, I interchange x and y, which reverses the domain and range between the function and its inverse.
90. I used vertical lines to determine if my graph represents a one-to-one function.
In Exercises 91–94, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
91. The inverse of is .
92. The function is one-to-one.
93. If , then .
94. The domain of f is the same as the range of .
95. If and , find and .
96. Show that
is its own inverse.
97. Freedom 7 was the spacecraft that carried the first American into space in 1961. Total flight time was 15 minutes and the spacecraft reached a maximum height of 116 miles. Consider a function, s, that expresses Freedom 7’s height, s(t), in miles, after t minutes. Is s a one-to-one function? Explain your answer.
98. If , and f is one-to-one, find x satisfying .
Group Exercise
99. In Tom Stoppard’s play Arcadia, the characters dream and talk about mathematics, including ideas involving graphing, composite functions, symmetry, and lack of symmetry in things that are tangled, mysterious, and unpredictable. Group members should read the play. Present a report on the ideas discussed by the characters that are related to concepts that we studied in this chapter. Bring in a copy of the play and read appropriate excerpts.
101. The size of a television screen refers to the length of its diagonal. If the length of an HDTV screen is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch? (Section P.8, Example 8)
102. Solve and graph the solution set on a number line:
Exercises 103–105 will help you prepare for the material covered in the next section.
103. Let and . Find . Express the answer in simplified radical form.
104. Use a rectangular coordinate system to graph the circle with center and radius 1.
105. Solve by completing the square: .
1.8: Exercise Set
1.8 Exercise Set
Practice Exercises
In Exercises 1–10, find and and determine whether each pair of functions f and g are inverses of each other.
1. and
2. and
3. and
4. and
5. and
6. and
7. and
8. and
9. and
10. and
The functions in Exercises 11–28 are all one-to-one. For each function,
Find an equation for , the inverse function.
Verify that your equation is correct by showing that and .
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
Which graphs in Exercises 29–34 represent functions that have inverse functions?
29.
30.
31.
32.
33.
34.
In Exercises 35–38, use the graph of f to draw the graph of its inverse function.
35.
36.
37.
38.
In Exercises 39–52,
Find an equation for .
Graph f and in the same rectangular coordinate system.
Use interval notation to give the domain and the range of f and .
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
(Hint for Exercises 49–52: To solve for a variable involving an nth root, raise both sides of the equation to the nth power: .)
49.
50.
51.
52.
Practice PLUS
In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function.
x
1
0
4
1
5
2
x
g(x)
0
1
1
4
2
10
53.
54.
55.
56.
57.
58.
In Exercises 59–64, let
Evaluate the indicated function without finding an equation for the function.
59.
60.
61.
62.
63.
64.
Application Exercises
Way to Go Holland was the first country to establish an official bicycle policy. It currently has over 12,000 miles of paths and lanes exclusively for bicycles. The graph shows the percentage of travel by bike and by car in Holland, as well as in four other selected countries. Use the information in the graph to solve Exercises 65–66.
Consider a function, f, whose domain is the set of the five countries shown in the graph. Let the range be the percentage of travel by bike in each of the respective countries. Write the function f as a set of ordered pairs.
Write the relation that is the inverse of f as a set of ordered pairs. Is this relation a function? Explain your answer.
66.
Consider a function, f, whose domain is the set of the five countries shown in the graph. Let the range be the percentage of travel by car in each of the respective countries. Write the function f as a set of ordered pairs.
Write the relation that is the inverse of f as a set of ordered pairs. Is this relation a function? Explain your answer.
67. The graph represents the probability of two people in the same room sharing a birthday as a function of the number of people in the room. Call the function f.
Describe in practical terms the meaning of , and .
68. A study of 900 working women in Texas showed that their feelings changed throughout the day. As the graph indicates, the women felt better as time passed, except for a blip (that’s slang for relative maximum) at lunchtime.
Source: D. Kahneman et al., “A Survey Method for Characterizing Daily Life Experience,” Science
Does the graph have an inverse that is a function? Explain your answer.
Identify two or more times of day when the average happiness level is 3. Express your answers as ordered pairs.
Do the ordered pairs in part (b) indicate that the graph represents a one-to-one function? Explain your answer.
69. The formula
is used to convert from x degrees Celsius to y degrees Fahrenheit. The formula
is used to convert from x degrees Fahrenheit to y degrees Celsius. Show that f and g are inverse functions.
Explaining the Concepts
70. Explain how to determine if two functions are inverses of each other.
71. Describe how to find the inverse of a one-to-one function.
72. What is the horizontal line test and what does it indicate?
73. Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.
74. How can a graphing utility be used to visually determine if two functions are inverses of each other?
75. What explanations can you offer for the trends shown by the graph in Exercise 68?
Technology Exercises
In Exercises 76–83, use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one).
76.
77.
78.
79.
80.
81.
82.
83.
In Exercises 84–86, use a graphing utility to graph f and g in the same by viewing rectangle. In addition, graph the line and visually determine if f and g are inverses.
84.
85.
86.
Critical Thinking Exercises
Make Sense? In Exercises 87–90, determine whether each statement makes sense or does not make sense, and explain your reasoning.
87. I found the inverse of in my head: The reverse of multiplying by 5 and subtracting 4 is adding 4 and dividing by 5, so .
88. I’m working with the linear function and I do not need to find in order to determine the value of .
89. When finding the inverse of a function, I interchange x and y, which reverses the domain and range between the function and its inverse.
90. I used vertical lines to determine if my graph represents a one-to-one function.
In Exercises 91–94, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
91. The inverse of is .
92. The function is one-to-one.
93. If , then .
94. The domain of f is the same as the range of .
95. If and , find and .
96. Show that
is its own inverse.
97. Freedom 7 was the spacecraft that carried the first American into space in 1961. Total flight time was 15 minutes and the spacecraft reached a maximum height of 116 miles. Consider a function, s, that expresses Freedom 7’s height, s(t), in miles, after t minutes. Is s a one-to-one function? Explain your answer.
98. If , and f is one-to-one, find x satisfying .
Group Exercise
99. In Tom Stoppard’s play Arcadia, the characters dream and talk about mathematics, including ideas involving graphing, composite functions, symmetry, and lack of symmetry in things that are tangled, mysterious, and unpredictable. Group members should read the play. Present a report on the ideas discussed by the characters that are related to concepts that we studied in this chapter. Bring in a copy of the play and read appropriate excerpts.
101. The size of a television screen refers to the length of its diagonal. If the length of an HDTV screen is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch? (Section P.8, Example 8)
102. Solve and graph the solution set on a number line:
Exercises 103–105 will help you prepare for the material covered in the next section.
103. Let and . Find . Express the answer in simplified radical form.
104. Use a rectangular coordinate system to graph the circle with center and radius 1.
105. Solve by completing the square: .
1.8: Exercise Set
1.8 Exercise Set
Practice Exercises
In Exercises 1–10, find and and determine whether each pair of functions f and g are inverses of each other.
1. and
2. and
3. and
4. and
5. and
6. and
7. and
8. and
9. and
10. and
The functions in Exercises 11–28 are all one-to-one. For each function,
Find an equation for , the inverse function.
Verify that your equation is correct by showing that and .
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
Which graphs in Exercises 29–34 represent functions that have inverse functions?
29.
30.
31.
32.
33.
34.
In Exercises 35–38, use the graph of f to draw the graph of its inverse function.
35.
36.
37.
38.
In Exercises 39–52,
Find an equation for .
Graph f and in the same rectangular coordinate system.
Use interval notation to give the domain and the range of f and .
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
(Hint for Exercises 49–52: To solve for a variable involving an nth root, raise both sides of the equation to the nth power: .)
49.
50.
51.
52.
Practice PLUS
In Exercises 53–58, f and g are defined by the following tables. Use the tables to evaluate each composite function.
x
1
0
4
1
5
2
x
g(x)
0
1
1
4
2
10
53.
54.
55.
56.
57.
58.
In Exercises 59–64, let
Evaluate the indicated function without finding an equation for the function.
59.
60.
61.
62.
63.
64.
Application Exercises
Way to Go Holland was the first country to establish an official bicycle policy. It currently has over 12,000 miles of paths and lanes exclusively for bicycles. The graph shows the percentage of travel by bike and by car in Holland, as well as in four other selected countries. Use the information in the graph to solve Exercises 65–66.
Consider a function, f, whose domain is the set of the five countries shown in the graph. Let the range be the percentage of travel by bike in each of the respective countries. Write the function f as a set of ordered pairs.
Write the relation that is the inverse of f as a set of ordered pairs. Is this relation a function? Explain your answer.
66.
Consider a function, f, whose domain is the set of the five countries shown in the graph. Let the range be the percentage of travel by car in each of the respective countries. Write the function f as a set of ordered pairs.
Write the relation that is the inverse of f as a set of ordered pairs. Is this relation a function? Explain your answer.
67. The graph represents the probability of two people in the same room sharing a birthday as a function of the number of people in the room. Call the function f.
Describe in practical terms the meaning of , and .
68. A study of 900 working women in Texas showed that their feelings changed throughout the day. As the graph indicates, the women felt better as time passed, except for a blip (that’s slang for relative maximum) at lunchtime.
Source: D. Kahneman et al., “A Survey Method for Characterizing Daily Life Experience,” Science
Does the graph have an inverse that is a function? Explain your answer.
Identify two or more times of day when the average happiness level is 3. Express your answers as ordered pairs.
Do the ordered pairs in part (b) indicate that the graph represents a one-to-one function? Explain your answer.
69. The formula
is used to convert from x degrees Celsius to y degrees Fahrenheit. The formula
is used to convert from x degrees Fahrenheit to y degrees Celsius. Show that f and g are inverse functions.
Explaining the Concepts
70. Explain how to determine if two functions are inverses of each other.
71. Describe how to find the inverse of a one-to-one function.
72. What is the horizontal line test and what does it indicate?
73. Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.
74. How can a graphing utility be used to visually determine if two functions are inverses of each other?
75. What explanations can you offer for the trends shown by the graph in Exercise 68?
Technology Exercises
In Exercises 76–83, use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one).
76.
77.
78.
79.
80.
81.
82.
83.
In Exercises 84–86, use a graphing utility to graph f and g in the same by viewing rectangle. In addition, graph the line and visually determine if f and g are inverses.
84.
85.
86.
Critical Thinking Exercises
Make Sense? In Exercises 87–90, determine whether each statement makes sense or does not make sense, and explain your reasoning.
87. I found the inverse of in my head: The reverse of multiplying by 5 and subtracting 4 is adding 4 and dividing by 5, so .
88. I’m working with the linear function and I do not need to find in order to determine the value of .
89. When finding the inverse of a function, I interchange x and y, which reverses the domain and range between the function and its inverse.
90. I used vertical lines to determine if my graph represents a one-to-one function.
In Exercises 91–94, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
91. The inverse of is .
92. The function is one-to-one.
93. If , then .
94. The domain of f is the same as the range of .
95. If and , find and .
96. Show that
is its own inverse.
97. Freedom 7 was the spacecraft that carried the first American into space in 1961. Total flight time was 15 minutes and the spacecraft reached a maximum height of 116 miles. Consider a function, s, that expresses Freedom 7’s height, s(t), in miles, after t minutes. Is s a one-to-one function? Explain your answer.
98. If , and f is one-to-one, find x satisfying .
Group Exercise
99. In Tom Stoppard’s play Arcadia, the characters dream and talk about mathematics, including ideas involving graphing, composite functions, symmetry, and lack of symmetry in things that are tangled, mysterious, and unpredictable. Group members should read the play. Present a report on the ideas discussed by the characters that are related to concepts that we studied in this chapter. Bring in a copy of the play and read appropriate excerpts.
101. The size of a television screen refers to the length of its diagonal. If the length of an HDTV screen is 28 inches and its width is 15.7 inches, what is the size of the screen to the nearest inch? (Section P.8, Example 8)
102. Solve and graph the solution set on a number line:
Exercises 103–105 will help you prepare for the material covered in the next section.
103. Let and . Find . Express the answer in simplified radical form.
104. Use a rectangular coordinate system to graph the circle with center and radius 1.
105. Solve by completing the square: .
Objective 1: Find the distance between two points
The Distance Formula
Objective 1 Find the distance between two points.
Watch Video
Using the Pythagorean Theorem, we can find the distance between the two points and in the rectangular coordinate system. The two points are illustrated in Figure 1.77.
The distance that we need to find is represented by d and shown in blue. Notice that the distance between the two points on the dashed horizontal line is the absolute value of the difference between the x-coordinates of the points. This distance, , is shown in pink. Similarly, the distance between the two points on the dashed vertical line is the absolute value of the difference between the y-coordinates of the points. This distance, , is also shown in pink.
Because the dashed lines are horizontal and vertical, a right triangle is formed. Thus, we can use the Pythagorean Theorem to find the distance d. Squaring the lengths of the triangle’s sides results in positive numbers, so absolute value notation is not necessary.
This result is called the distance formula.
The Distance Formula
The distance, d, between the points and in the rectangular coordinate system is
To compute the distance between two points, find the square of the difference between the x-coordinates plus the square of the difference between the y-coordinates. The principal square root of this sum is the distance.
When using the distance formula, it does not matter which point you call and which you call .
Example 1 Using the Distance Formula
Find the distance between and . Express the answer in simplified radical form and then round to two decimal places.
Find the distance between and . Express the answer in simplified radical form and then round to two decimal places.
Objective 1: Find the distance between two points
The Distance Formula
Objective 1 Find the distance between two points.
Watch Video
Using the Pythagorean Theorem, we can find the distance between the two points and in the rectangular coordinate system. The two points are illustrated in Figure 1.77.
The distance that we need to find is represented by d and shown in blue. Notice that the distance between the two points on the dashed horizontal line is the absolute value of the difference between the x-coordinates of the points. This distance, , is shown in pink. Similarly, the distance between the two points on the dashed vertical line is the absolute value of the difference between the y-coordinates of the points. This distance, , is also shown in pink.
Because the dashed lines are horizontal and vertical, a right triangle is formed. Thus, we can use the Pythagorean Theorem to find the distance d. Squaring the lengths of the triangle’s sides results in positive numbers, so absolute value notation is not necessary.
This result is called the distance formula.
The Distance Formula
The distance, d, between the points and in the rectangular coordinate system is
To compute the distance between two points, find the square of the difference between the x-coordinates plus the square of the difference between the y-coordinates. The principal square root of this sum is the distance.
When using the distance formula, it does not matter which point you call and which you call .
Example 1 Using the Distance Formula
Find the distance between and . Express the answer in simplified radical form and then round to two decimal places.
Find the distance between and . Express the answer in simplified radical form and then round to two decimal places.
Objective 2: Find the midpoint of a line segment
The Midpoint Formula
Objective 2 Find the midpoint of a line segment.
Watch Video
The distance formula can be used to derive a formula for finding the midpoint of a line segment between two given points. The formula is given as follows:
The Midpoint Formula
Consider a line segment whose endpoints are and . The coordinates of the segment’s midpoint are
To find the midpoint, take the average of the two x-coordinates and the average of the two y-coordinates.
Example 2 Using the Midpoint Formula
Find the midpoint of the line segment with endpoints and .
Solution
To find the coordinates of the midpoint, we average the coordinates of the endpoints.
Find the midpoint of the line segment with endpoints (1, 2) and .
Circles
Our goal is to translate a circle’s geometric definition into an equation. We begin with this geometric definition.
Definition of a Circle
A circle is the set of all points in a plane that are equidistant from a fixed point, called the center. The fixed distance from the circle’s center to any point on the circle is called the radius.
Figure 1.80 is our starting point for obtaining a circle’s equation. We’ve placed the circle into a rectangular coordinate system. The circle’s center is and its radius is r. We let represent the coordinates of any point on the circle.
Figure 1.80 A circle centered at with radius r
What does the geometric definition of a circle tell us about the point in Figure 1.80? The point is on the circle if and only if its distance from the center is r. We can use the distance formula to express this idea algebraically:
Squaring both sides of yields the standard form of the equation of a circle.
Objective 3: Write the standard form of a circle’s equation
Objective 3 Write the standard form of a circle’s equation.
Watch Video
The Standard Form of the Equation of a Circle
The standard form of the equation of a circle with center and radius r is
Example 3 Finding the Standard Form of a Circle’s Equation
Write the standard form of the equation of the circle with center (0, 0) and radius 2. Graph the circle.
Solution
The center is (0, 0). Because the center is represented as (h, k) in the standard form of the equation, and . The radius is 2, so we will let in the equation.
The standard form of the equation of the circle is . Figure 1.81 shows the graph.
Write the standard form of the equation of the circle with center (0, 0) and radius 4.
Example 3 and Check Point 3 involved circles centered at the origin. The standard form of the equation of all such circles is , where r is the circle’s radius. Now, let’s consider a circle whose center is not at the origin.
Example 4 Finding the Standard Form of a Circle’s Equation
Write the standard form of the equation of the circle with center and radius 4.
Solution
The center is . Because the center is represented as in the standard form of the equation, and . The radius is 4, so we will let in the equation.
The standard form of the equation of the circle is .
Write the standard form of the equation of the circle with center and radius 10.
Objective 4: Give the center and radius of a circle whose equation is in standard form
Objective 4 Give the center and radius of a circle whose equation is in standard form.
Watch Video
Example 5 Using the Standard Form of a Circle’s Equation to Graph the Circle
Find the center and radius of the circle whose equation is
Graph the equation.
Use the graph to identify the relation’s domain and range.
Solution
We begin by finding the circle’s center, (h, k), and its radius, r. We can find the values for h, k, and r by comparing the given equation to the standard form of the equation of a circle, .
We see that , and . Thus, the circle has center and a radius of 3 units.
To graph this circle, first locate the center . Because the radius is 3, you can locate at least four points on the circle by going out 3 units to the right, to the left, up, and down from the center.
The points 3 units to the right and to the left of are and , respectively. The points 3 units up and down from are and , respectively.
Using these points, we obtain the graph in Figure 1.82.
The four points that we located on the circle can be used to determine the relation’s domain and range. The points and show that values of x extend from to 5, inclusive:
The points and show that values of y extend from to , inclusive:
Find the center and radius of the circle whose equation is
Graph the equation.
Use the graph to identify the relation’s domain and range.
If we square and in the standard form of the equation in Example 5, we obtain another form for the circle’s equation.
This result, , suggests that an equation in the form can represent a circle. This is called the general form of the equation of a circle.
The General Form of the Equation of a Circle
The general form of the equation of a circle is
where D, E, and F are real numbers.
Objective 4: Give the center and radius of a circle whose equation is in standard form
Objective 4 Give the center and radius of a circle whose equation is in standard form.
Watch Video
Example 5 Using the Standard Form of a Circle’s Equation to Graph the Circle
Find the center and radius of the circle whose equation is
Graph the equation.
Use the graph to identify the relation’s domain and range.
Solution
We begin by finding the circle’s center, (h, k), and its radius, r. We can find the values for h, k, and r by comparing the given equation to the standard form of the equation of a circle, .
We see that , and . Thus, the circle has center and a radius of 3 units.
To graph this circle, first locate the center . Because the radius is 3, you can locate at least four points on the circle by going out 3 units to the right, to the left, up, and down from the center.
The points 3 units to the right and to the left of are and , respectively. The points 3 units up and down from are and , respectively.
Using these points, we obtain the graph in Figure 1.82.
The four points that we located on the circle can be used to determine the relation’s domain and range. The points and show that values of x extend from to 5, inclusive:
The points and show that values of y extend from to , inclusive:
Find the center and radius of the circle whose equation is
Graph the equation.
Use the graph to identify the relation’s domain and range.
If we square and in the standard form of the equation in Example 5, we obtain another form for the circle’s equation.
This result, , suggests that an equation in the form can represent a circle. This is called the general form of the equation of a circle.
The General Form of the Equation of a Circle
The general form of the equation of a circle is
where D, E, and F are real numbers.
Objective 5: Convert the general form of a circle’s equation to standard form
Objective 5 Convert the general form of a circle’s equation to standard form.
Watch Video
We can convert the general form of the equation of a circle to the standard form . We do so by completing the square on x and y. Let’s see how this is done.
Example 6 Converting the General Form of a Circle’s Equation to Standard Form and Graphing the Circle
Write in standard form and graph: .
Solution
Because we plan to complete the square on both x and y, let’s rearrange the terms so that x-terms are arranged in descending order, y-terms are arranged in descending order, and the constant term appears on the right.
This last equation, , is in standard form. We can identify the circle’s center and radius by comparing this equation to the standard form of the equation of a circle, .
In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
1. (2, 3) and (14, 8)
2. (5, 1) and (8, 5)
3. and
4. and
5. (0, 0) and
6. (0, 0) and
7. and
8. and
9. and (4, 1)
10. and (4, 3)
11. (3.5, 8.2) and
12. (2.6, 1.3) and
13. and
14. and
15. and
16. and
17. and
18. and
In Exercises 19–30, find the midpoint of each line segment with the given endpoints.
19. (6, 8) and (2, 4)
20. (10, 4) and (2, 6)
21. and
22. and
23. and
24. and
25. and
26. and
27. and
28. and
29. and
30. and
In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius.
31. Center (0, 0),
32. Center (0, 0),
33. Center (3, 2),
34. Center
35. Center
36. Center
37. Center
38. Center
39. Center
40. Center
In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation’s domain and range.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
Practice PLUS
In Exercises 65–66, a line segment through the center of each circle intersects the circle at the points shown.
Find the coordinates of the circle’s center.
Find the radius of the circle.
Use your answers from parts (a) and (b) to write the standard form of the circle’s equation.
65.
66.
In Exercises 67–70, graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations.
67.
68.
69.
70.
Application Exercises
The smartphone screen shows coordinates of six cities from a rectangular coordinate system placed on North America by long-distance telephone companies. Each unit in this system represents mile.
Source: Peter H. Dana
In Exercises 71–72, use this information to find the distance, to the nearest mile, between each pair of cities.
71. Boston and San Francisco
72. New Orleans and Houston
73. A rectangular coordinate system with coordinates in miles is placed with the origin at the center of Los Angeles. The figure indicates that the University of Southern California is located 2.4 miles west and 2.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake’s epicenter is estimated to be approximately 30 miles from the university. Write the standard form of the equation for the set of points that could be the epicenter of the quake.
74. The Ferris wheel in the figure has a radius of 68 feet. The clearance between the wheel and the ground is 14 feet. The rectangular coordinate system shown has its origin on the ground directly below the center of the wheel. Use the coordinate system to write the equation of the circular wheel.
Explaining the Concepts
75. In your own words, describe how to find the distance between two points in the rectangular coordinate system.
76. In your own words, describe how to find the midpoint of a line segment if its endpoints are known.
77. What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.
78. Give an example of a circle’s equation in standard form. Describe how to find the center and radius for this circle.
79. How is the standard form of a circle’s equation obtained from its general form?
80. Does represent the equation of a circle? If not, describe the graph of this equation.
81. Does represent the equation of a circle? What sort of set is the graph of this equation?
82. Write and solve a problem about the flying time between a pair of cities shown on the smartphone screen for Exercises 71–72. Do not use the pairs in Exercise 71 or Exercise 72. Begin by determining a reasonable average speed, in miles per hour, for a jet flying between the cities.
Technology Exercises
In Exercises 83–85, use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window.
83.
84.
85.
Critical Thinking Exercises
Make Sense? In Exercises 86–89, determine whether each statement makes sense or does not make sense, and explain your reasoning.
86. I’ve noticed that in mathematics, one topic often leads logically to a new topic:
87. To avoid sign errors when finding h and k, I place parentheses around the numbers that follow the subtraction signs in a circle’s equation.
88. I used the equation to identify the circle’s center and radius.
89. My graph of is my graph of translated 2 units right and 1 unit down.
In Exercises 90–93, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
90. The equation of the circle whose center is at the origin with radius 16 is .
91. The graph of is a circle with radius 6 centered at .
92. The graph of is a circle with radius 5 centered at .
93. The graph of is a circle with radius 6 centered at .
94. Show that the points , and are collinear (lie along a straight line) by showing that the distance from A to B plus the distance from B to C equals the distance from A to C.
95. Prove the midpoint formula by using the following procedure.
Show that the distance between and is equal to the distance between and .
Use the procedure from Exercise 94 and the distances from part (a) to show that the points , , and are collinear.
96. Find the area of the donut-shaped region bounded by the graphs of and .
97. A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is at the point .
Retaining the Concepts
98. Determine whether the graph of is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
Exercises 101–103 will help you prepare for the material covered in the next section.
101. Write an algebraic expression for the fare increase if a $200 plane ticket is increased to x dollars.
102. Find the perimeter and the area of each rectangle with the given dimensions:
40 yards by 30 yards
50 yards by 20 yards.
103. Solve for h: . Then rewrite in terms of .
1.9: Exercise Set
1.9 Exercise Set
Practice Exercises
In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
1. (2, 3) and (14, 8)
2. (5, 1) and (8, 5)
3. and
4. and
5. (0, 0) and
6. (0, 0) and
7. and
8. and
9. and (4, 1)
10. and (4, 3)
11. (3.5, 8.2) and
12. (2.6, 1.3) and
13. and
14. and
15. and
16. and
17. and
18. and
In Exercises 19–30, find the midpoint of each line segment with the given endpoints.
19. (6, 8) and (2, 4)
20. (10, 4) and (2, 6)
21. and
22. and
23. and
24. and
25. and
26. and
27. and
28. and
29. and
30. and
In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius.
31. Center (0, 0),
32. Center (0, 0),
33. Center (3, 2),
34. Center
35. Center
36. Center
37. Center
38. Center
39. Center
40. Center
In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation’s domain and range.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
Practice PLUS
In Exercises 65–66, a line segment through the center of each circle intersects the circle at the points shown.
Find the coordinates of the circle’s center.
Find the radius of the circle.
Use your answers from parts (a) and (b) to write the standard form of the circle’s equation.
65.
66.
In Exercises 67–70, graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations.
67.
68.
69.
70.
Application Exercises
The smartphone screen shows coordinates of six cities from a rectangular coordinate system placed on North America by long-distance telephone companies. Each unit in this system represents mile.
Source: Peter H. Dana
In Exercises 71–72, use this information to find the distance, to the nearest mile, between each pair of cities.
71. Boston and San Francisco
72. New Orleans and Houston
73. A rectangular coordinate system with coordinates in miles is placed with the origin at the center of Los Angeles. The figure indicates that the University of Southern California is located 2.4 miles west and 2.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake’s epicenter is estimated to be approximately 30 miles from the university. Write the standard form of the equation for the set of points that could be the epicenter of the quake.
74. The Ferris wheel in the figure has a radius of 68 feet. The clearance between the wheel and the ground is 14 feet. The rectangular coordinate system shown has its origin on the ground directly below the center of the wheel. Use the coordinate system to write the equation of the circular wheel.
Explaining the Concepts
75. In your own words, describe how to find the distance between two points in the rectangular coordinate system.
76. In your own words, describe how to find the midpoint of a line segment if its endpoints are known.
77. What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.
78. Give an example of a circle’s equation in standard form. Describe how to find the center and radius for this circle.
79. How is the standard form of a circle’s equation obtained from its general form?
80. Does represent the equation of a circle? If not, describe the graph of this equation.
81. Does represent the equation of a circle? What sort of set is the graph of this equation?
82. Write and solve a problem about the flying time between a pair of cities shown on the smartphone screen for Exercises 71–72. Do not use the pairs in Exercise 71 or Exercise 72. Begin by determining a reasonable average speed, in miles per hour, for a jet flying between the cities.
Technology Exercises
In Exercises 83–85, use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window.
83.
84.
85.
Critical Thinking Exercises
Make Sense? In Exercises 86–89, determine whether each statement makes sense or does not make sense, and explain your reasoning.
86. I’ve noticed that in mathematics, one topic often leads logically to a new topic:
87. To avoid sign errors when finding h and k, I place parentheses around the numbers that follow the subtraction signs in a circle’s equation.
88. I used the equation to identify the circle’s center and radius.
89. My graph of is my graph of translated 2 units right and 1 unit down.
In Exercises 90–93, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
90. The equation of the circle whose center is at the origin with radius 16 is .
91. The graph of is a circle with radius 6 centered at .
92. The graph of is a circle with radius 5 centered at .
93. The graph of is a circle with radius 6 centered at .
94. Show that the points , and are collinear (lie along a straight line) by showing that the distance from A to B plus the distance from B to C equals the distance from A to C.
95. Prove the midpoint formula by using the following procedure.
Show that the distance between and is equal to the distance between and .
Use the procedure from Exercise 94 and the distances from part (a) to show that the points , , and are collinear.
96. Find the area of the donut-shaped region bounded by the graphs of and .
97. A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is at the point .
Retaining the Concepts
98. Determine whether the graph of is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
Exercises 101–103 will help you prepare for the material covered in the next section.
101. Write an algebraic expression for the fare increase if a $200 plane ticket is increased to x dollars.
102. Find the perimeter and the area of each rectangle with the given dimensions:
40 yards by 30 yards
50 yards by 20 yards.
103. Solve for h: . Then rewrite in terms of .
1.9: Exercise Set
1.9 Exercise Set
Practice Exercises
In Exercises 1–18, find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places.
1. (2, 3) and (14, 8)
2. (5, 1) and (8, 5)
3. and
4. and
5. (0, 0) and
6. (0, 0) and
7. and
8. and
9. and (4, 1)
10. and (4, 3)
11. (3.5, 8.2) and
12. (2.6, 1.3) and
13. and
14. and
15. and
16. and
17. and
18. and
In Exercises 19–30, find the midpoint of each line segment with the given endpoints.
19. (6, 8) and (2, 4)
20. (10, 4) and (2, 6)
21. and
22. and
23. and
24. and
25. and
26. and
27. and
28. and
29. and
30. and
In Exercises 31–40, write the standard form of the equation of the circle with the given center and radius.
31. Center (0, 0),
32. Center (0, 0),
33. Center (3, 2),
34. Center
35. Center
36. Center
37. Center
38. Center
39. Center
40. Center
In Exercises 41–52, give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation’s domain and range.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
Practice PLUS
In Exercises 65–66, a line segment through the center of each circle intersects the circle at the points shown.
Find the coordinates of the circle’s center.
Find the radius of the circle.
Use your answers from parts (a) and (b) to write the standard form of the circle’s equation.
65.
66.
In Exercises 67–70, graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations.
67.
68.
69.
70.
Application Exercises
The smartphone screen shows coordinates of six cities from a rectangular coordinate system placed on North America by long-distance telephone companies. Each unit in this system represents mile.
Source: Peter H. Dana
In Exercises 71–72, use this information to find the distance, to the nearest mile, between each pair of cities.
71. Boston and San Francisco
72. New Orleans and Houston
73. A rectangular coordinate system with coordinates in miles is placed with the origin at the center of Los Angeles. The figure indicates that the University of Southern California is located 2.4 miles west and 2.7 miles south of central Los Angeles. A seismograph on the campus shows that a small earthquake occurred. The quake’s epicenter is estimated to be approximately 30 miles from the university. Write the standard form of the equation for the set of points that could be the epicenter of the quake.
74. The Ferris wheel in the figure has a radius of 68 feet. The clearance between the wheel and the ground is 14 feet. The rectangular coordinate system shown has its origin on the ground directly below the center of the wheel. Use the coordinate system to write the equation of the circular wheel.
Explaining the Concepts
75. In your own words, describe how to find the distance between two points in the rectangular coordinate system.
76. In your own words, describe how to find the midpoint of a line segment if its endpoints are known.
77. What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.
78. Give an example of a circle’s equation in standard form. Describe how to find the center and radius for this circle.
79. How is the standard form of a circle’s equation obtained from its general form?
80. Does represent the equation of a circle? If not, describe the graph of this equation.
81. Does represent the equation of a circle? What sort of set is the graph of this equation?
82. Write and solve a problem about the flying time between a pair of cities shown on the smartphone screen for Exercises 71–72. Do not use the pairs in Exercise 71 or Exercise 72. Begin by determining a reasonable average speed, in miles per hour, for a jet flying between the cities.
Technology Exercises
In Exercises 83–85, use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window.
83.
84.
85.
Critical Thinking Exercises
Make Sense? In Exercises 86–89, determine whether each statement makes sense or does not make sense, and explain your reasoning.
86. I’ve noticed that in mathematics, one topic often leads logically to a new topic:
87. To avoid sign errors when finding h and k, I place parentheses around the numbers that follow the subtraction signs in a circle’s equation.
88. I used the equation to identify the circle’s center and radius.
89. My graph of is my graph of translated 2 units right and 1 unit down.
In Exercises 90–93, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
90. The equation of the circle whose center is at the origin with radius 16 is .
91. The graph of is a circle with radius 6 centered at .
92. The graph of is a circle with radius 5 centered at .
93. The graph of is a circle with radius 6 centered at .
94. Show that the points , and are collinear (lie along a straight line) by showing that the distance from A to B plus the distance from B to C equals the distance from A to C.
95. Prove the midpoint formula by using the following procedure.
Show that the distance between and is equal to the distance between and .
Use the procedure from Exercise 94 and the distances from part (a) to show that the points , , and are collinear.
96. Find the area of the donut-shaped region bounded by the graphs of and .
97. A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is at the point .
Retaining the Concepts
98. Determine whether the graph of is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.
